While it would improve immersion, I do think there’s a few flaws with the idea.

Sepermeru & Asagarth trade goods, so they aren’t actively hostile to one another.
A Relic Hunter outfit would work here to enter Asagarth peacefully.

The Summoning Place on the other hand is a rather exclusive club… a fair skinned Hyborian is probably going to arouse more than a little suspicion wearing Darfari garb.

The Lemurians are a relatively small and (I imagine) tight-knit group being refugees. Sneaking in, even with a disguise, would be difficult.

I can see it working for pirate and exiles camps easily though, either group will take anyone in. The same goes for the Dogs of the Desert, more or less.