'поменьше' rather than just 'меньше' means 'a little smaller,' not just 'smaller'
Yes, in some cases поменьше is a weaker form of меньше, meaning a bit smaller. Another meaning would be smaller than current one, and this is the case here. In this particular sentence меньше sounds unnatural, though probably technically correct. The following variants are much better:
У вас есть рубашка меньшего размера?
У вас есть рубашка чуть меньше?
У вас есть рубашка меньше, чем эта?
English English (as oppose to, perhaps, American English) does not require the word "got" in, for instance, "Have you a smaller shirt?". It is so maddening to be told you have made a mistake because you omitted a redundant word!
Maybe you should word it as "Do you have" instead of "Have you." That might help.
Yes, it does. (Not saying that it is an accurate translation, I am not sure about that. But the sentence makes sense in English.)