What with the component situation what it currently is, this may not be the best time to be experimenting too much. That said, most of what is said, and everyone "knows" about load variations as a result of primer changes are worth just as much as most other "common knowledge". They are mostly what has been repeated over and over by gun writers for decades and are based more on "it sounds right to me" rather than actual evidence based on experimentation. This goes for what is said on the subject in all reloading manuals too.
The only way to know what will happen with a particular powder with different primers is to pick a load, any cartridge, bullet and powder, but all the same with only the primer being changed. .357 in an "N" frame is a good place to start, using a charge weight which is known to be below "maximum" by at least 5%. Actually a 158 LSWC with 13.5 gr. 2400 would be an excellent load for this purpose. Load at least 10 rounds with each of several different primers, standard, magnum, and small rifle, and different brands of each, depending on how comprehemsive a comparison you wish to make. 20 rounds or more will result in a more statistically significant sampling, the more the better even as many as 100 or more of each.
Fire your loads over your chronograph on the same day, within a reasonable time so the temperature is constant within a few degrees, and from the same gun. If you have chronographed several samples of a particular load previously you will have a good starting point for comparing the differences you might observe with everything being apparently equal but from different samples. It is not uncommon to see average velocity vary by 20-30 FPS or more between 10 round samples, even if fired from the same batch of ammunition. ES and SD can vary also.
Now, considering all the information you have as a base, shoot your loads, all with each primer in one string. When done compare Mean (average) Velocity, ES and SD for all the different primers. When this is done, and remembering that pressure equates to velocity, and the converse, see what the results were. I can confidently predict you will be surprised by what you find. "Common Knowledge" says the magnum primers will give higher pressure (velocity) than standard, and the small rifle even higher than the magnum. 'Taint necessarily so. What you will find is the standard will give higher, lower or virtually identical results to both the magnum and small rifle primers. SD and ES may be higher or lower too. Remember, that any difference as a result of the primer must be greater than the difference of identical samples fired at either different times or even consecutively for the difference to indicate a significant difference in any parameter.
Other things you may see that don't appear to be logical are that the ES and SD are higher/lower between samples, and that this parameter does not equate to the mean velocity. Higher mean velocity may have a lower or higher ES/SD.
Hopefully this makes some sense to you. Remember also that the result is only entirely valid with that powder and lot, and the specific lot number of each different primer. This will not mean that a particular primer type and a different powder will always give similar results.
How do I know this? I am not guessing. Over the past 20 years I have shot literally thousands of rounds over my Oehler 35P. I have made just this comparison quite a few times with several cartridges. I have usually been comparing two specific primers such as SM and Small Rifle, but have done LP/LPM too. Sometimes I have gotten lower velocity with a SPM compared to SP, but higher ES and SD. Sometimes velocity has been higher. This goes for SR/SP or SR/SPM comarisons too. There were variations in those parameters which are usually taken to indicate the consistency of a load, the ES and SD, but I have NEVER seen what could be considered as a statistically significant variation in velocity.
Just one other small point about primers. Notice above that I have tested Small Rifle primers in several revolver cartridges, mostly .357 Magnum. Even though I have been a practicing gunsmith/police and military armorer for many years I do not tinker with the mainspring in any of my revolvers, there just isn't any reason to in most cases. With stock springs I have literally NEVER had a mis-fire as a result of using a small rifle primer in a revolver cartridge. This included several thousand rounds of .32-20 in several S&W and Colt revolvers, all primer brands, and more years than many of you have been alive. Leave the mainspring alone and the strain screw tight and rarely will you have a mis-fire which can be blamed on a "Hard" primer.