Any Small Rifle Primer works (I don't follow all the names, I assume they are small rifle). So yes, you can substitute Fiachi or any other small rifle primer.
The only aspect to be careful with is if you are at the high end of the load.
Different mfg primers have a different affect on pressure. Its not huge but it is (or can be there)
So, back off a high end load. Or if the powder you are using has a narrow load range.
Personally my loads range from really low (7.5 Swiss) to about half way between medium and max (6.5 Lapua). The low load I would do nothing for.
The Lapua as its not pushing max load, I would also do nothing with, I would load some test rounds (10-15, see where they are velocity wise and accuracy wise (assuming I don't have any pressure indicators). If I have any pressure indicators I would back off half a grain. Pull the bullets and test again.
I have run comparison test in large rifle in 40-06 between CCI and Remington and I see no statistical difference in velocity. Velocity is your hallmark for pressure. You are not going to increase velocity unless the pressure goes up.
It does help to have a Chrono though you can also look for pressure signs in case rub (back of Rim) sticky bolt and most unreliable but always worth a look at the primer (pierced, cratered with a rim)
I use all the indicators and any one raising its head is enough to make me drop back.