The .540 bolt face isn't going to "get-er done" for a .375 ELR rig.
There are a few options and I'd say the .590 bolt face (37XC or a necked down 416 Rigby / 378 Weatherby) is going to be the smallest bolt face for the 3000 yard cartridge that your specifying.
A .630 bolt face is used for Chey-Tac Brass
So you need to decide how much "long green" your willing to spend.
I'm not a ELR steel plate shooter competitor, but I do shoot E2K (Groups & Score)
I know that the Chey-Tac case has a good amount of ELR rifles built on it and has done well in the standings of some of the matches where I had an opportunity to get to see published results.
- I know that there are ELR matches held at several locations (here is a link for schedule:
https://elrcentral.com/elr-matches/)
If you really want to pursue ELR my advise is to attend some events and connect with those who are active in the discipline. - I'm sure there is a Hugh wealth of knowledge in those guys and you will learn plenty from them.