Zhong Shan Suit Inner Shirt Cantonese Name
CantoDict doesn't contain a word like「底」, and given that the character「」is located in the CJK Unified Ideographs Extension B block, it's not a common use character. An appropriate reading and meaning for「」appears in the Kangxi Dictionary:



However, that definition says「衣短袖」, which is almost the exact opposite to「長袖衫」. If whoever you heard it from is not mistaken, then that will be a Cantonese dialect word, and not Standard Cantonese.

The closest entry in CantoDict which matches the meaning is「底衫」(dai2 saam1, undershirt). I'm going to note here that「長袖衫」doesn't actually mean inner shirt, it means long-sleeved shirt, and if this tailor that you're seeing designs inner shirts with short sleeves instead then「底」(or maybe「底衫」) sounds entirely appropriate.