Tell us about yourself!
I believe what you are detecting when you hold your Geiger counter near an intact (closed) smoke alarm, are the low intensity gamma rays emitted by the Americium. The alpha particles are stopped inside. The hazard would be if Americium particles were dislodged from the metal base while it is being disassembled. I had a A.C. Gilbert spinthariscope, 60 years ago, with radium in it. I'm still here. Sunlight causes cancer with great certainty. This project, probably will not, if it is done carefully.
This detector is well worth what it cost to build it, if you have a consuming interest in it, or a significant need for it. For a learner, and experimenter, the parts bill is pretty high, due to the single unit price on the PIN diode detectors. The BPW-34 you mention, is inexpensive, but not very sensitive. Is there a "middle ground" device, with better performance, but produced in commodity volume, so the price is more affordable?
I think your "compressor converted into a vacuum pump" is a good solution. My only problem is the noise and vibration from those little pumps is annoying. Do you think a 12Vdc "Windshield Washer Fluid" pump would take too much time? I do not mean pumping the fluid through the pump (although it is rated for alcohol). I mean using the ability of the pump to lift fluid on its intake side (by vacuum). Most would have 50-60 (cm of H2O) lift ability on the intake side.