Umm..since the op hasn't responded since April, one might think there was no real problem, and/or no 'frozen' rifle.
The longest common primers are the W209, hence using W209 primer provides the best chance for a 'seal', without modification. Use of the RIO primer just might require one to shim the firing pin bushing out some, as you suspect. If i understand correctly what you mean, the thickness of the flange isn't too important; what determines the seal is the total length of the primer.
The RIO primer probably won't reliably ignite Blackhorn , unless you also have the 'made for Blackhorn breech plug' coming along with the rifle.
Myself, i see no reason to be nervous about having a small amount of 'crush' of the primers, after you install the shims. If there isn't a small amount of crush.. how will there be a seal against blow back? Another option is to use a 1mm thick o-ring between the primer and the breech plug.. This would be done without shimming the firing pin bushing, and may require the use of primers as short as the STS. Typically the STS are a few thousandths shorter, than other commonly available primers. Having never seen a RIO primer, i have no idea how long they are, or if they will fit with the o-ring in place.
At any rate, a small amount of 'crush', i.e. 0.002", does not create a dangerous condition; closing the action will not cause the primer to 'pop'. Rifle brands other than CVA, have been 'crushing' the longer primers for years, without mishap.