I suspect the O.P. specifically wants an incandescent bulb rotating beam beacon, presumably for its visual effect, possibly as an advertising display. A xenon or LED strobe beacon has a very different appearance, and many people find the harsh omnidirectional flash visually distracting.
The problem with retrofitting a LED bulb is they are very unlikely to have a uniform light output with respect to viewing angle, so the visual effect of using one with a rotating reflector will be significantly different. Also it may be a PITA to fit anything other than a
H1 lamp. Its possible modifying it to use a vertically mounted LED spot bulb and replacing the rotating reflector with a 45° mirror to 'throw' the spot beam horizontally would produce the desired rotating beam effect. Some experimentation would be advisable e.g. with a mirror on a turntable and a gooseneck spotlight centred over it, before hacking up the existing rotating beacon.
Availability of H1 24V 55W halogen bulbs as I suggested earlier isn't great, as most truckers prefer 70W or even 100W bulbs but they aren't too hard to find online. Higher voltage and lower power non-LED bulbs are rare like hens teeth, though 35W H1 HID bulbs and 24V-36V 35V HID ballasts are available if you don't mind the price.
Floobydust has a good point about not simply upping the voltage to compensate for voltage drop in the wiring, with the 12V bulb directly in parallel to the motor due to the risk of motor burnout if the bulb fails, but if the motor is powered from a buck regulator, as I proposed, that objection goes away. It would be helpful if the O.P. could measure the motor current @12V with the bulb removed.