Unless you have additional information to the contrary, you should be able to nurse her to meet her demands. Your supply will increaae with more frequent nursing. It can be deceptive because our breasts feel so much less full at night than in the morning, but odds are there is enough for her.
That being said, I do like to give my daughter a bottle occasionally for convenience. Sometimes, I'm out and just don't want to nurse in publis. Sometimes, like at the playground, we are both hot and tired, and it is easiest to sit her in the stroller and give her a cold bottle to cool her off and not have to get all sweaty holding her. So I think it is worthwhile trying it yourself occasionally, as long as it is not at the expense of your supply. I can usually get away with giving 2-3 bottles per week. But if I am doing it more frequently, then I definately pump.
I personally find it very awkward to give a bottle since it feels so different than nursing, so I imagine it feels the same way for her. Try to feed your baby in a very different position. I frequenly put my baby in her stroller and face her or sit her upright against my chest. So you might try a different position.
Also, I think that you could introduce a sippy cup. It might be a bit early, but she should be pretty competent with it by around 6 months.
My daughter is also four months old and is my second child. I learned the first time around, and am realizing this time to, that I often mistakenly think my child is hungry when the exact opposite is true. A breastfed baby often wants that non-nutritive sucking on the breast just to relax and is often annoyed if there is milk coming out. It comes across as fussing and hunger. Similarly, when babies teeth or have ear infections, it is the same. It is actually pain not hunger. So if your child is acting strange at the breast, it may be that she is NOT hungry, and that is why she is rejecting the bottle form you (or maybe just playing with it). She might actually want YOU but not the MILK.
So, please don't worry about your supply, unless you know for a fact you don't produce much. Babies are great regulators of their mom's production. If you do have supply issues, see a lactation consultant because they can work wonders, and they can actually measure how much of a feed they are getting in ounces. And finally look for cues that she may not be hungry. Try just rocking, giving a pacifier, or putting her down (some babies get overstimulated and just want to be in their cribs!)
Hope this helps.