Would a lawyer by any other name still speak bullshit? Yes. But why are we talking about lawyers and judges?

Because I think that in the proposed scenario, where people hire master rationalists to arbitrate disputes, these arbitrators and other rationalists who would be hired by each side independently for advice would start behaving like judges and lawyers do, respectively. (Although case law probably wouldn't become important.)

You have explained well the reason that capping is a terrible idea. Now it is time to update the 'probably capped' part.

I didn't mean they would be capped by guild rules or something like that, but rather, that the effective market prices would stay low. I've no proof of this, economics is not my strong suit, but here are the reasons I think that's likely to happen:

1. If arbitration is so expensive that some people can't afford it (and it needs to be affordable by the poorer party in a conflict), that's an untapped market someone could profit from. Whenever your argument is with a poor party, you have to have an arbitrator whose fee is at most twice the fee the other party can or is willing to afford, and there's no effective low limit here. (State-provided judges and loser pays winner's fees do have something going for them.)
2. Being a better arbitrator doesn't require direct investment of money on part of the arbitrator. So a good arbitrator who's not getting enough work can lower his prices.
3. Third parties interested in seeing a dispute resolved - if only to achieve peace and unity - might contribute money towards the fee, or send volunteer arbitrators, in exchange for the parties to the dispute agreeing to arbitration. Finally, competition between arbitrators (for money and work) would eventually draw the fees down, assuming a reasonable supply of arbitrators.
4. What would make people choose an arbitrator that wasn't the cheapest available? Assuming some kind of minimal standard or accreditation (e.g., LW karma > 1000), an arbitrator is inferior if he cannot properly comprehend your rational argument or might be swayed by your opponent's master rationalist lawyer. You then have a choice: invest your money in a costlier and fairer arbitrator, or in a better lawyer so you can sway the cheap arbitrator to your own side. I do hope that one dollar buys more unswayingness than swaying-power, but with humans you never know.
5. How does someone prove she's a good arbitrator in the first place? Wouldn't you need another, more senior arbitrator to decide on that and to handle appeals? Either there's a hierarchy, in which case the lowest ranked arbitrators are cheap (because it's their own entry price in the business); or there's a less centralized web of trust, and if it's fragmented enough the whole idea of universally trusted arbitration is undermined.