To make a performance of the poem perfectly regular, all you have to do is not pause after "all the king's men." Instead, go galloping off into the last line, not stressing "Couldn't" but rather stressing "put," just as you would do in normal speech. If the beats fall on "put," "Dump-," "-get-," and "-gain," equal time will exist between all beats in all lines, with only four beats per line.
The reason the line gives trouble is that it is the only perfectly regular line in the poem. In all the earlier lines, the two opening unstressed syllables have been deleted. Here they are not.