You should have no problem forging with a beckett burner, as long as you have good diesel fuel, as far as I understand the main reason that very few people do this is the sheer price of diesel fuel, but if this isn’t a concern then this should work well for you. And as far as oxidation goes you shouldn’t have any more problems than with a coal forge.
Here is a link to a thread that may or may not help :-)
https://www.bladesmithsforum.com/index.php?/topic/21694-oil-burner-forge/