Rechargeable cells should work OK. However they only provide 1.2V each, rather than the 1.5V that 'normal' batteries do. When compared to such 'normal' batteries, they're starting from a lower voltage, so they might not last as long.
As long as you're using alkaline batteries though (and not the dreaded zinc carbon ones), a set of 'normal' batteries should last quite a long time anyway. Mouse manufacturers like to say 'up to a year' but that must be for very light usage; I reckon on perhaps two to three months for a pair of batteries.
As you can buy 36 good quality AAA alkaline batteries for £8 in B&M, that's only about 45p per pair, which I reckon isn't bad for several months use:
https://www.bmstores.co.uk/products/panasonic-aaa-batteries-10-8-free-322533