Does it need to be specifically a "Champ"? ....and are you looking for clean, distorted, or edge of breakup sounds?
The way I usually play a tweed amp, is with all controls dimed, then I use the volume on my guitar and touch sensitivity to alter the dirt, and the tone controls or pickup selector on the guitar to tweak the tone.
I just profiled my 1955 Tremolux (original speaker, transformers, etc) at full and half volume through a 1073 preamp and an Oktava 319 mic, and could try to upload the profiles if you're interested (I haven't made any uploads yet, so I've got to figure out how).
The '55 Tremolux is the first model, and is dirtier (think tweed Deluxe) than the newer ones due to the design being a little different. It might be too dirty for you at full volume, but at half volume it's really touch sensitive and at either setting you can roll back the volume on your guitar and get a pretty cool edge of breakup tone that could be useful for jazz/blues.
Whether you like the tone will be up to your ears, but I can guarantee you that the profile sounds like the amp (warts and all) and has a healthy amount of speaker breakup and character as you'd expect from a 60+ year old original.
I use the same approach with my '66 Deluxe that I uploaded to the Rig Exchange a few years ago. The amp was turned up to "7" when I profiled it. I use the guitar controls and playing dynamics to go from clean and sparkly to gainy, dark lead. It is called "Fender Dyn DR paults"
