What emulsifier can be used for an Egg-free mayonnaise

I think the problem is actually not the heating, but the refrigeration!

Consider: bacon fat solidifies easily and thickly, and does so even at room temperature. When you throw it in the fridge, the micro-droplets of bacon fat will turn to solids and clump together. These solid droplets are frozen in place when solid, but when you thaw it, they melt and reveal how wrecked your emulsion is. Now, you could add additional emulsifying agents (lecithin, extra yolks) but that's not really going to solve the problem of refrigeration.

The solution is to break up the bacon fat as it melts, and re-establish the emulsion before it can break. To do this, you have to whisk constantly as you gently heat the mayonnaise, generally in a warm water bath.

How to execute this in a restaurant setting:

Prep a big batch of bacon-mayo and throw it in the fridge. Just before service, warm some water in a pot and throw a cup of the mayo in a small bain marie or metal 6th hotel pan. Immerse the bain/pan in the pot, and whisk it as it melts. Hold the mayonnaise for service in water warm enough to melt bacon fat, and DISCARD THE WARM MAYONNAISE EVERY TWO HOURS AND THAW A FRESH BATCH. Make sure no cooks get lazy about that -- it's a food safety problem. Ideally, you should be using pasteurized eggs to reduce the risk of salmonella. Basically, you're treating your mayonnaise like a Hollandaise or Bearnaise.

I've found that the thawing trick works fairly well for home hollandaise (which most people say shouldn't refrigerate), and which does the same thing if heated fast. It should apply to easily-broken mayonnaise too.

Yes, it will work. The texture won't be different but the flavor will be significantly different. It will taste like lime hollandaise. :-P

I imagine a lime hollandaise would taste great on fish and other seafood. I'm not entirely sure what you would do with lime mayonnaise though.