http://www.hebrewworks.com/Transliteration.htm Since the transliterations produced with this program are customizable, I was wondering if this program counts as a source for the existence of the spellings it can create. For example, I can use it to write בֵית as "bhait," but I'm wondering if it counts as a valid source for that since I could also use it to write בֵית as "ffudj."
As long as you're mapping symbols in one script to symbols in another script, it counts as a transliteration. For example, 05D105B505D905EA is also a valid transliteration of בֵית, as is байт, or n'ɔ̤. So yes, anything you code into SAFFA counts as a transliteration. It just might not be a particularly useful one.
The question you should be asking instead is, is this transliteration useful? And that depends on the purpose you want to use it for. 05D105B505D905EA is a reasonable way to transcribe בֵית if, say, you're writing a program and the programming language doesn't support non-ASCII characters. But if I sent that to a Hebrew-speaking friend, they would probably just be confused. n'ɔ̤ might be useful if I somehow had an IPA keyboard but not a Hebrew one, but isn't at all useful for anyone who can't already read Hebrew script (since it's just imitating the shape of the Hebrew letters).
If this is just for your own personal use (which I'm guessing it is, based on the previous iterations of this question), and you find it useful to write בּ as bh, then go for it; in my personal notes on Egyptian, for example, I use non-standard letters like 9, ç, and x—because they're easier to type than the standard ones, and I can read them just fine. They work for me, and therefore they're useful for the purpose of my notes.
I guess the answer is a plain no to the question. A customisable program can create a lot of misleading, invalid, or wrong spellings as "transliterations", therefore none of the spellings can be quoted as existent, legitimate, or canonical unless you stick to a documented and accepted transliteration scheme. In this case, the transliteration scheme should be quoted with a reference and the program is only a tool to implement it, doing the transliteration by hand following the scheme should give exactly the same result.