2006 Honda Element blown engine

Bad day. Aborted trip to Anchorage. Mystery oil leak developed. Oil warning light did not come on.
Four quarts oil lost. Now blowing smoke bad and lost power. Rings and who knows what else.

Any suggestions on a good rebuilt engine manufacturer?

You might try to find an engine from a wrecked vehicle in a salvage yard if you can. This will probably be a lot cheaper and the yard should be able to tell you what engines will work.

there seem to be a couple of sources of used/rebuilt engines for this car you could use Car-part.com and search through what’s available for your area from various salvage yards (with #of miles on engine) but your Honda mechanic could know a trustworthy source