Get rid.
Wait til Liz dies, since she does have a lot of goodwill across the UK and since she'll most likely be dead in less than a decade, then don't crown another.
Turn Buckingham Palace into an even more gigantic moneyspinner by opening it up like Verseilles (or the Winter Palace in St Petersburg, or the Topkapi Palace in Istanbul, or lots of famous palaces in former monarchies). People visit for the architecture, history, art, and culture associated with the Palaces - not the living bodies that are sometimes resident inside.
Write down the bloody constitution for the first time ever so there's a complete and single source of constitutional law in the UK. In it, specify the rules and qualifications necessary to become president (disbar former prime ministers? no problem) and clearly delineate powers so the UK has a proper system of democratic checks and balances for the first time ever.
Breathe a sigh of relief.
Funny how lots of monarchists seem to think they're winding up republicans by saying they want to keep the monarchy. Hardly - it's not exactly a surprise that some people like Queenie. We just think you're wrong 