It is likely an instance of "swamp-speak": 'essentially a rural southern U.S. dialect laced with nonstop malapropisms, fractured grammar, "creative" spelling and mangled polysyllables such as "incredibobble" and "hysteriwockle," plus invented words such as the exasperated exclamations "Bazz Fazz!," "Rowrbazzle!" and "Moomph!"' (according to Wikipedia).
Pogo is set in the Okefenokee swamp, but it is written by an Irish-American from the Northeast, and many strips are social and/or political satire. The language is part of the art of the strip and uses lots of wordplay like puns, poetry, malapropisms, and literary allusions. So much of that strip is dependent on what was going on in the world at the time a particular strip was written, that to fully understand it you might need to look at American newspapers from that month in history to see what was going on.
I have found some evidence that "tilly hawk" is another name for a sparrow-hawk, but I don't think that really fits with the context. It could be an allusion to the name of someone or some group of note during the 1948 elections that the author is mocking. Clay pigeons are used in skeet shooting, so given what Alan found in his answer, those clay targets could be what tilly bird alludes to. This is complete speculation - as FumbleFingers mentioned in the comments, it isn't a meaningful phrase outside of that one strip, so I personally wouldn't dig too deep trying to understand it exactly.
I would interpret it as the opposition getting beaten so badly it's like they didn't put up a fight, i.e. it was like shooting fish in a barrel.