The IMR handloaders guide, formerly supplied by Du Pont, recommends the Remington 6 1/2 primer for handloading the .22 Jet cartridge.
My NRA Handloading manual identifies the primer size for the .22 Remington Jet cartridge as "small rifle," and again shows the Remington 6 1/2 primer being used in most handloads.
The maximum product average chamber pressure for this cartridge, according to the NRA handloading manual, is 42,100 c.u.p.
I have seen other handloading data using small pistol primers in the .22 Remington Jet cartridge. Those loads are often assembled with reduced powder charges and, ideally, less pressure.
If you intend to load full power ammunition you may find the Remington 6 1/2 primer a good choice, at least to start with.
Good luck with your project and keep us posted.