If someone wanted to basecamp on the Canadian side of a border lake, what would they need for a permit?

Look in the Quetico forum here at the second pin down. Everything you need is there.
Quetico forum

If you camp on the Canadian side of the border you have entered Canada. Regardless of how far you go across the border you are subject to a host of rules and restrictions that are totally different than staying in the USA in the BW.
You would each need a Remote Area Border Crossing permit, subject to passing their criminal background check. About $30. Has to be done a couple of months in advance.
You would need a Q permit at about $21 bucks a day per person, so 2 days each if you camp overnight.
You would need a passport or card to re-enter the US and will have to use the new ROAM app to check back into the US. Free
And, of course, you would be subject to various park and customs rules for Canada and the Q.
Other than that it is easy.

GC - we pay for number of nights so two days/one night is one day fee.

The harder part is you have to pick up the permit from a ranger station the day of it day before you enter. Requires planning. It could work picking up a permit at PP for Carp or Basswood River.

There is a "Agnes Lake/Basswood Basecamp" permit that allows you to camp on the Canada side of Basswood or enter Agnes, but you are not supposed to use this to camp on any of the other lakes between Basswood and Agnes (Sunday for example) without going to Agnes first.