Re:
Is writing to the cloud lookup tables the most reliable action from the bp? I am assuming we would have far less potential disruptions from bp > lookup table as opposed to bp > web request?
I can't speak to "reliability" in a statistical sense there, but a Business Process > Lookup Table call doesn't have external dependencies, which seems important here.
I agree that an "explicitly confirm receipt before proceeding" flow like you outlined is the way to go. That could be the "shared table" mechanism, or something simpler like Laserfiche only treating specific HTTP response codes as success and retrying/not proceeding until it gets an HTTP 200 back.
If the business objective is "don't advance the process until we're 100% confident the other side has received and processed the request to avoid complex manual resubmission", you need a mechanism by which the other side can affirmatively tell Laserfiche "I received and consumed the request, you're safe to proceed."
Trying to confirm web requests sends on the Laserfiche side feels like an XY Problem. That's not to say the try/catch mechanism I outlined for sends isn't still worthwhile for reliability/error handling, but a confirmed send isn't an actionable signal for the business logic here. Only a confirmed consumed receive is, so I'd focus on the confirmation mechanism.