Zappo
Explorer
Pity I came in late. I think Gallo22 is more right than not, and the player is more wrong than not. However, I don't believe that following the rules is a limit for a good DM, and for this reason I don't really agree on ignoring the rules for the sake of challenge, story, or whatever, when it isn't needed.
What I would have done is having the bridge be a literal rope bridge (one rope to walk on, two ropes to hang onto), or a narrow plank without ropes. DC 20 is within the rules for this, and it's a challenge. Saying "yeah this looks just like a DC 15 bridge, except that it's DC 20" may be a challenge, but actually justifying the higher DC is better.
What I would have done is having the bridge be a literal rope bridge (one rope to walk on, two ropes to hang onto), or a narrow plank without ropes. DC 20 is within the rules for this, and it's a challenge. Saying "yeah this looks just like a DC 15 bridge, except that it's DC 20" may be a challenge, but actually justifying the higher DC is better.