Show the Mobile Safari Debug Console on demand
No.

You can only turn it on or off, there is no way to set it only to certain sites.

Instead of using the (limited) debug console on the iphone, try using Safari on a desktop and set the user agent to iPhone (From in the Developer Menu, to enable the developer menu, open Safari preferences and choose Advanced where there is a checkbox to enable it).

You should then be able to catch almost all of the console logs, as Safari and MobileSafari are naturally similar. Although there are things you will need to use the iPhone for (like viewport issues)