Mouse rating 1.5V | Can I use a 1.2V rechargeable battery safely?

Your 1.2V rechargeable battery should work perfectly well.

An alkaline battery starts of with a terminal voltage somewhere between 1.5 - 1.6 Vdc which decays as the battery is discharged. End of life is a terminal voltage somewhere between 0.7 - 1.0 Vdc (depends on the equipment using the battery).

Your 1.2V rechargeable cell starts off with a terminal voltage around 1.2 Vdc which remains fairly constant as the cell discharges. Only when the cell charge is exhausted does the terminal voltage drop significantly.