Two Verbal Job Offers/RLMT

I currently have two verbal jobs offers. They have explained that they need to conduct the RLMT before they can proceed with other formalities (i.e., issuing an official job offer letter).

I've explicitly said yes (but only verbally through an agency) to the first employer and implicitly said yes to the second by agreeing for them to conduct the RLMT. However, I know that I will ultimately have to pick one or the other. I'm trying to figure out if I'm being prudent or screwing up my reputation by keeping both of them on the hook (or perhaps I'm being both). Should I inform one of them ASAP that I've picked the other or is it unwise to do that prior to the RLMT?

My other issue with this is that in previous job searches, I've always been told to never take an offer seriously until you get it officially, in writing, and obviously neither employer is legally able to do that until RLMT is done. Should I just wait for both of them to conduct RLMT and then pick one shortly before they get to the CoS step? Obviously I wouldn't want two companies to attempt to file CoS for me.

I've been assured by both recruiters that RLMT shouldn't be a problem, but I am also anxious about just believing verbal promises. I mean, technically there's no guarantee they won't come out of RLMT with a different person they want to hire, right?

The fact that the first one is through an agency is also a pain communication-wise, because I feel like I'm playing a game of telephone (or BrE 'Chinese whispers'), where I am always the last person in the loop, possibly with the most distorted information. But the second company also seems new to sponsorship, so who knows which one is better to trust.

Any ideas here? Should I tell one employer to stop RLMT and go with the other?

If I was you I would work in my own interest.