Instead of muddling lime and sugar to make a Caipirinha, could I make a lime syrup? Or even just add a simple syrup to some lime juice?

I would try making a lime syrup and add water if needed. If you do it the other way, you might find that it takes a lot of lime juice to get the flavor you want and get more than you need. Also, you could try adding some lime zest for the muddled look and added flavor without adding acidity. Try blanching the zest in boiling water and shock it in ice water. This will make it a vibrant green, and take out the undesirable flavor of the peel.