Three moves is probably in the realm where you could manually brute-force it, although it may take some effort.
Pick a move, a reply, a second move, and a second reply. Then start looking at whether White has a checkmate from that position. If not, then White's second move was incorrect, so change it. If you eliminate all possible White second moves this way, then go back to White's first move, and change it. If you eliminate all White first moves too, then you have proven that a mate in 3 does not exist.
If you do find a White checkmate from one of these positions, then change Black's previous move. If the solution is valid, White will have a checkmate for all possible Black replies.