S.P.
There are a lot of opinions here! You seem to have touched on something that many people have feelings and thoughts about. I'll add mine to the bundle - I think my views are in the minority, but they come from my long (18 years) as a mom whose daughters have a step-mom and my personal experience AS a step-mother.
First, PICK YOUR BATTLES. Let go of issues like bikinis. The truth is that older children need permission of ALL parents (and step-parents) to do something, not just one. Any other way would mean that the child who is told "NO" by one parent will simply go looking for another parent who will tell them "YES."
Second, while it is true that the birth mother has a giant stake in what happens to her daughter, it is also a bad and awkward thing for the ex-wife to have power or control over what happens inside your home - especially THROUGH your boyfriend. If that becomes the process, then you will find yourself arguing and begging for your side of things, and she will be doing the same. That puts your boyfriend in a very uncomfortable middle spot, and it puts you in a very weak spot - not as a partner inside your own household, but as someone who has to beg your boyfriend to be on your side. First and foremost, you and your boyfriend have moved in together and are now a "team" and should decide together what is right for your family, in your household. He will still have to negotiate with his ex about their mutual children, but he should not agree to anything different than what you've decided without first saying to her, "I need to talk to K. about this and see what she thinks," or, if relations are friendly enough, "why don't we talk to K. about this and see what thoughts she has?" His ex needs to understand that your needs and thoughts are part of the equation, just like hers are.
Here are some things to consider:
1. If you were the regular babysitter, the child's mom would speak to you directly about how you care for her child. If you are going to have that babysitter role, consider the idea that you and your boyfriend ask his ex to discuss child care matters with you directly, just as she would with any paid caretaker. Remember, too, that while you are "getting something" out of this (the ability to stay home with your own child), she is also getting something - free child care!
2. I think no outsider should have a right to tell you whether or not you may have a dog. But they do have a right to take their child out of your home if they don't want her to be around a dog. Go back to the paid caretaker scenario and think about how this would work. You can understand why a mom would be upset about her child being bit. Anyone leaving their child with you might react by saying they do not want the dog in the vicinity. It might or might not be an over-reaction (was it puppy rambunctiousness or a "mean dog"? Was it playful teething, or dislike of children?) (Bassets are generally pretty mellow, so I'm guessing this is puppiness). Anyway, this could be the response of any parent. Then, there would be certain choices that had to be made. You could offer up to her that you would keep the dog away from the child (caged or in another, closed room) and get the dog obedience training. At that point, she can accept that solution or not. If she does not accept that solution, then you would have to decide whether you want the dog enough to lose the "business" (e.g. her taking her child to a different caretaker), AND she would have to decide whether she wanted the hassle of changing caretakers enough to pull the child out of your home and find someone else. That is how it would work in a paid care situation.
3. We haven't talked at all yet about whether the dog is really the issue, or whether the real issue is the fact that the ex's new lady is the caretaker of her child. As much as you may wish this were a neutral situation, there are probably uncomfortable feelings there - the woman who has replaced her is now "parenting" her child all day. And, you get to stay home with HER child, while she either has to or wants to work. Even if she wants to, there may be a part of her that is jealous. Her trying to control things in your household may be stemming from some underlying resentment. In the end, it might not be a great idea to babysit the ex's daughter by another man. If you feel you MUST babysit someone else's child in order to justify staying home with your own child, consider taking in a paying customer or two. That way, you will have what is called an "arms' length" relationship - a business relationship. People can choose your home based on who YOU are, and what sort of home you have. They can choose to leave the care situation, if they don't like it - that's just business. You can fire customers who you don't like, too! The situation you are in now could have so many negative ramifications, especially with your husband stuck in the middle. You will have enough possible issues through the years, because they have a child in common. But you and your boyfriend might not necessarily want the added burden of having to argue over a child that is not his.
ONE LAST PLUG for choosing your battles (I am too wordy, I know). You, your boyfriend AND his ex should consider which battles are critical. During their lives, children will have many caretakers, and even with good care, none will care for them in the exact way that mom does. The amazing thing is that children are resilent, and can benefit and adjust from having different care takers. For example, kids will have several different teachers over the course of their lives, yet they manage to learn from year to year new rules, new procedures, etc, and they still thrive and cope. As long as you are a good, safe, loving parent, children will do fine around you. Mostly you can work out differences in parenting styles by talking them through, but occasionally, the parent should just relax and let things be done differently - if it won't hurt anything. Here is an example of where I do not agree with EITHER of the parents, yet it is ok for me to "do it my way":
In our household, we have the kids every other week. On our weeks, our routine is that I go downstairs and fix breakfast for everyone, while the Dad wakes the three kids up and gets them started dressing. Then he goes to shower, etc, and get ready for work. The oldest boy wants highly sugared cereal for breakfast. His mom and his dad both are ok with that, but I am a healthy eater, and it drives me nuts that the first food of the day is loaded with sugar - like dessert! I know many moms don't think this is bad, especially since these cereals generally have vitamins added, etc. However, it bothers ME - think me weird or whatever. But you know what? If I'm going to be the one who feeds them, and I'm the one who shops, and I'm the one who has to watch them eat it, then I need to be able to make a decision I am comfortable with. When this came up, I offered to the Dad that HE take over the breakfast job himself, and I will get them out of bed and get them started dressing. That way, if he wanted to give them sugary cereal I wouldn't have to watch. Of course, that would have meant him getting up half and hour earlier so that he could shower first, and he decided that his 1/2 hour of sleep was more important than feeding his son sugary cereal.
Here you see, I am not trying to TELL HIM how to parent. I am just saying that this way of parenting doesn't work for me, and I cannot comfortably parent that way. If he wants to parent that way, FINE. I won't try to stop him.
I am not always this stubborn. There are other things, like bedtime, whether the kids need to eat their vegatables, how often they bathe and brush their teeth, that we have had disagreements about. Usually we compromise and work something out that we can both live with. Sometimes I give in and do it his way, sometimes he gives in and does it my way.
I don't know if any of this has been helpful, K., but I want to emphasize that the first chore you have is to make sure you and your boyfriend are on the same page. Otherwise, you will find yourself very frustrated, and an UNequal partner in your own home who has to get permission.
Step-parenting is very very difficult. Good luck with it!
S.