Although some of the first video projectors used conventional filament lamps, yours will have a discharge or metal halide type of lamp. The price of the replacement lamp you stated would be about right for an discharge lamp.
It's not a straightfoward swap to put a halogen lamp in a projector. The power supply for the discharge lamp is current regulated, between 25 to 140 volts depending on the lamp. The lamp igniter produces a high voltage ignition pulse which would cause damage to the old lamp socket if no arc lamp is installed, and would blow up if connected across a kitchen spotlight bulb.
Halogen lamps produce much more infrared than a metal halide lamp of comparable size, and this will damage the polarising filters in the optical assembly.
Obtaining even illumination of the whole picture is more difficult than you may think. The colour temperature of the lamps is different, incandescent are yellow, and arc lamps are more like daylight, this will affect the picture.
Lastly the projector electonics monitors the lamp status, and shuts down if there is an error.
It's not an option unfortunately. Nice idea, but you may ruin a perfectly good projector trying.