Success! After two trips through a hot washer and a hot dryer, the Monmouth cap shrank to a fraction of its former size, became very thick, and the Cotswold bloomed beautifully, as I had hoped. It took two trips because after the first "felting cycle," Stephen pronounced it to be still a bit too large (it came down over his ears too much). As you can see in the photographs, there's still some indication that the cap was knitted, but it's beginning to get very fuzzy.
One problem I discovered during the first felting cycle: do not felt fuzzy knits with terrycloth towels. I spent about 15 minutes carefully picking little black lint balls off the hat. As they were a different color black than the black in the hat, I knew they must have come from the black terrycloth towel I threw in the washer with the cap to help with the felting. For the second felting cycle, the cap went into the washer alone, and there were no more cotton lint balls.
After the second felting cycle, the hat was smaller, thicker, and the individual stitches in the knitting obscured by the fuzziness of the Cotswold bloom. Stephen tried it on and it gave him slightly goofy, just-fell-off-the-haywagon look that I suspect all these knitted and felted caps gave their wearers. I'm planning on teaseling (gently brushing the surface to bring up the nap) and then shearing the nap to improve the water-shedding ability these caps supposedly had. I would love to use actual teasels--they grow in vacant lots in the Bay Area--but this is the wrong season for them, so the dog brush I use for flicking open locks for spinning will have to suffice.
This has been a good experiment in reproducing 16th century knitting. I designed a yarn with characteristics to achieve a specific finished product, knitted something out of my handspun, achieved the level of felting I desired, and made Stephen something extremely cool that can't be purchased from a sutler or off the Internet.