I'd like to have dinner together, if I may.

Literal

Dinner [object-を] together-[humble-ご]-do-want-[explanatory-のですが].

御一緒する (ごいっしょする) is a humble-polite expression for 'joining' someone. The 御- (go-) prefix elevates 一緒 ('together'), and する turns the whole thing into a humble verb. This is the go-to polite way to propose spending time with someone — dining, going somewhere, etc. The trailing のですが softens it into a tentative offer.