These seem to be the way birthday events and invitations are set up in my network of friends and family, including wealthy and penniless, old and young, and different class cultures;
Friends or family set it up for the birthday person: “let’s take you out for your birthday “. Friends and family split bill and cover the cost of birthday person.
Birthday person issues invitation (typically to close family / offspring) “please join me for my birthday dinner”, and they may well pay for the event: the family members will know from custom and practice.
Someone holds a dinner or party in their house or a hall, or takes over a private room in a hotel ... generally the host will then pay, definitely if at home or a hall, but ‘bring a bottle’ is usual, and if a big occasion most people would bring a bottle of something bubbly anyway.
All these are perfectly acceptable, all that is needed is clarity, which the OP had achieved.
People are very short of money atm. I would much rather pay for a pub lunch and see my friend than her Dd be too embarrassed to organise it because she couldn’t afford to host everyone, or even worse, get into debt and struggle with bills.
OP: is the surprise element causing you problems? Suppose you just plan it with your Mum? And whatever you choose, surprise her with a cake?