J.R. Pierce, The Guru of the optimized Electron Gun from Bell Labs, is rolling in his grave.

I'll skip why the CRT will be amazingly difficult to bias correctly ,as you would need a few more electrodes and the cathode mount is "in the way", but may I suggest you build an "Omegatron", which with care, can be made with little more then a Dremel tool , diecast box, and some really good vacuum epoxy, plus a few old Cold War surplus metal Octal relays to salvage the glass to metal seals. That little 250 AMU Omegatron would be a good start and a lot easier.
PMTs have ion suppression as part of their design, the compounds on the electrodes will oxidize. Your looking for a "Channeltron" or Faraday Cup. As a chemist, you'd find Channeltrons are not going to need "Special technical Magic" to make.
See US Patent 3,152,280 and several open source NASA publications for details on Omegatrons.
Works well at higher pressures, and since there is no getter, you will not be fighting clouds of barium oxide powder jamming up your instrument, nor fighting the Aquadag coating when it massively outgasses every time you let it up to air. With an Omegatron, a FET electrometer is all you would need for detection and there is no voltage over one hundred volts, if even that much.
Not to mention the Watt or so of required tunable RF would be really easy these days to generate with a DDS board from Ebay.
Some of the "Time of Flight" strategies run at 1 ATM, no filament, and no more then 1500 volts with Mosfet pulsers and are simply made of Teflon and steel tubing. The commercial large molecule versions are handheld.
Dead quadrapole leak detectors are all over Ebay, and I miss the simple tower PC sized 1970s Veeco I used in a Lab.
If you did it with a CRT setup you'd have to dismount the gun, remove the cathode, add the ionizer, and put it in a new housing. That is, unless you like analyzing Ba-Ca-Sr carbonates, exactly once.
If you have access to "Review of Scientific Instruments, you'd find some really nice 1940s and 1960s MS designs that will get you there and consist of bent steel tubes for the housings.
Steve