AVI is just a container format, so whether or not you can burn it to a bluray without reencoding depends on the actual codec used. H.262, H.264, and VC-1 are supported without transcoding. H.265 or other codecs aren’t. If it’s a compatible codec, you can use something like tsMuxeR to get it into the proper m2ts format output it to blu ray without losing any quality.

You can also just try burning the AVI files to a Blu-ray data disk, as many players have a wide variety of formats that they can play from a BD-ROM disk, but check your player’s manual to see what they support.