Fixitt - if you tried to use untrimmed sheets, even though the solder might not stick to the channel walls, you would still have solder balls/clumps to get out of the block afterwards.
Here is an example of the solder paste I have been talking about for use - Solder-It Plumbers' Solder (CSP-41). Look at it
here. Several companies make similiar solder pastes, although the silver brazing paste is much more common.