I say go on the business trip. It will be good to be seen as a team player and will help you professionally. Being mommy tracked is not something you want to happen to you. Also I can almost guarantee your son won’t get markedly worse while you are away. It is a phase and it will carry on until he is ready for it to be done. Perhaps by leaving your husband to handle it alone, your son will actually be better. More than once my husband and I have noticed our kids are easier when the other parent is not around.
Our youngest has gone in and out of the anxiety thing as well. At almost two now he's settling back down again but it is tough while the phase lasts. First of all stop feeding the monster. Let him get all wound on his own. When our youngest starts in, my husband and I back off, ignore the shenanigans and carry on with the routine. Sometimes it helps if one of us takes charge, mushing him through the routine with the occasional "two minutes until we put on shoes", "two minutes until we leave"... The other person stays out of it until it is clear the other person needs help. Kids can't tell time so it's not like we really time it. We just give him a bit of advance warning. It could be one minute or ten minutes. The point is to prep for the inevitable.
Also never state things as questions if they aren’t really questions. My husband is notorious for saying, ‘Who’s ready to go to school?” When the answer from the kids is a resounding “no”, now what? Instead he has learned to say, “It’s time for school. Let’s go to the car.” Some things are choices and some are facts. Be clear because it is easier on the kids.
As for daycare drop off be quick, matter of fact and consistent. My husband takes our youngest to his classroom and sets him on the floor where he clings, cries and carries on. My husband gives him a quick kiss on the head, says “I love you and will see you this afternoon.” After repeating this over and over again, eventually the crying jags get shorter and shorter. His teacher now tells us for the most part he’s over the whole crying thing within five minutes.
At all times maintain a calm voice and calm approach. We have had some real firestorms in our house because toddlers are good at that. However, never once has a bad situation gotten better when we the adults cannot maintain our calm.
Good luck. This is just a phase no matter how exhausting it may seem.