Can a plan sponsor make a loan to a participant that the participant then uses to pay off a loan from the plan, or is this a prohibited transaction?

Is the plan sponsor making the loan from the plan? - that would certainly be prohibited. If making a loan from assets outside a qualified plan, I do not believe it would be prohibited.

If the loan from the sponsor is from the individual, and has nothing to do with the plan, and could be shown as a "business transaction" for the company books or the individuals accountant, then the plan isn't involved.