I have looked high and low for a running 350 and I'm telling ya, in SoCal they are few and far between...and if I do find them it's already been pulled and they want $1,500 for it.
Just for kicks, I just searched CL within a 250mi radius of LA (easy drive distance) and only saw a couple that looked promising. One for $200 in the car, hear it run scenario. But you're right, sbc engines seem scarce or expensive around you. Have you looked for a running vehicle that "needs" a bunch of work but has a decent engine? That's another avenue, albeit another challenge to trailer a junker to and from your house.
Based on your post above about LSing it anyways, I'd still be of the mindset to drop a running engine in it and get on with it. Different story if you were planning on keeping whatever engine goes back in "forever."
I would view it as spending undue time (re-building) on something that is not a sure thing anyway and while it may not cost more $, it surely won't save a bunch of $ but depends on how you value your time.
I made the mistake (maybe, based on my value of my time) of scraping the paint off the entire truck (I'm redoing the '86 currently) because it would peel, so it had to go. Seemed like it wouldn't "take that long". Well, I probably have, idk, 80 hours into scraping and sanding that damn truck and just now ready for bodywork and primer. I could have had it blasted for under $1000. Pull the bumpers, cover the windows and let er rip. OR I could have blasted it myself for almost free. Have 185cfm compressors I could borrow and the local rental houses would give me a sand pot for a weekend.
I thought I was "saving", but in the end, I could have had a truck, 95% prepped and ready for paint and body in a day or 1 weekend. The only upside of DIY the way I did it, is I could work on it for 30 minutes or 12 hours depending on how much time I have.