You can't. Some people just don't want pets because they don't want to care for them and they find them inconvenient when they want to travel or do things.

Bit much to expect your bf to forgo his birthday celebrations so your Mum can effectively have 2 days of celebration. You are an adult now, time to gain some independence. If you can't ever see yourself making your bf's birthday a priority over your Mum's, then just let him go. It's not fair of you to make it so that he will never be able to celebrate his birthday properly with yourself, his friends or even his family on his actual birthday.

Edit to add: he doesn't share a birthday with your Mum, her birthday is Jan 1st. So it's entirely reasonable that you don't see her at all on Dec 31.