In the world of Sales... the price you get is what someone is willing to pay. This willingness is based on many factors, but value is a big one. If you have something someone wants badly enough, the sales process is pretty straight forward especially if you are the only real option they have...you set the price, they pay.
If they are on the fence or just don't want it bad enough, then it's more of a process to convince them that they want it and should buy... Your job is to make their decision an easier one. Having barriers typically makes this value proposition more difficult for them to work out (Except in case as noted above where the barriers may not matter as much).
So, generally speaking, having the title= less barrier. (You will lose a lot of potential buyers simply because they will want a title and don't want to hassle with getting it or risk not being able to get it)
Having a running bike= a lot less barrier (you'll win someone who's on the fence simply by having them hear it run... they'll feel the roar, they'll visualize what they want: a killer bike for them to show off with... and a lot of the "Fixing" that it may need like a rusted front end, etc. are negated by the sound of the engine.
Without either of these as "Known" certainties, then you're in a position of "hoping" someone is willing to buy it. You don't have a lot of leverage and you're putting up barriers (Reasons) for them to not want to buy.
Another way to put this is You'll essentially need to convince them that A) it can be legally registered if THEY do the work and B) it can run in the future if THEY do the work...both of which means the VALUE has to be worth their time/interest...which equates to LESS $.
I don't know the current value of your BSA, but I do know the value of a sale (It's what I do)... and as much as I love BSA's, I am one of those that wouldn't consider your bike without both of those being addressed. I doubt I'm the minority when it comes to that...so unless you have a known buyer, I'd work on those two for the best chances of a higher dollar sale.
You may make more parting it out...but then how much is your time worth? No doubt you'll likely be spending a lot more of it advertising, stripping it down, and shipping it out...over a larger period of time. If cash is wanted sooner than later, getting the title probably is the biggest win for your time in my opinion. But the engine running may be a close second.
Hope that helps in some way... Good luck!