Optimal Cheese Melting Bagel Baking Temp/Time
To do this you want to apply heat to the cheese (to melt it) and not to the bagel (so it doesn't dry).

This is probably best achieved by grilling (broiling in the US) the bagel, rather than putting it in an oven, so the direct heat on the cheese melts it, but the bagel is not warmed so it doesn't crisp or dry.   

This assumes you have an open bagel obviously.  if you are have a closed bagel, with a top and a bottom, then I would melt the cheese on one half under the grill and the put the top piece on.

If you have to oven cook it then it's a trade off between high temp, which will crisp the bagel whilst the cheese is melting and the melting of the cheese should take a short time, or lower temp where the bagel is more likely to dry out as you need to leave it in for longer to get the melted cheese you want.