Yes. The two eventualities are equivalent for this purpose.
Obviously the non-existence of a solution might itself be unprovable, so you'd still have to set some time limit. Oh Gödel, always there to ruin our plans.
Obviously if no solution exists then that fact is also provable, since this is a finite problem. The problem is not that it might not be provable. The problem is that verifying a non-existence proof could be hard, because such a proof would probably be something like "I ran my solver for three weeks and it said that there are no solutions".
Obviously the non-existence of a solution might itself be unprovable, so you'd still have to set some time limit. Oh Gödel, always there to ruin our plans.