As near as I can tell, there are several methods for generating a single valid de Bruijn sequence using a given deterministic approach, as well as algorithms for churning out all possible sequences. For my purposes, I need only be able to generate random ones.
Ideal would be an algorithm where you can provide part of the sequence, and it will generate the rest of a valid sequence starting with that. But nearly as good would be any technique where I could provide a random seed or something equivalent and be likely to get a new distinct sequence each time I use it.
I'd like to be able to do this for, say, sequences of order-7 and higher, which is why any approach that requires iterating through all possible sequences up front is not feasible, unless it's computationally cheap to generate each next sequence and doesn't need to be done as a batch. If anybody can give me a helpful suggestion or useful link, I'll consider this answered.