I'm thinking 700 watts carrier or 2.8 KW PEP is too much drive on the 3CX3000 tube. If the box is put together right it will make over full rated power with far less drive. I used 400 watts carrier with 2 KW PEP drive on a 3CX6000A7 amp posted in the "tube amp pics" section. It produced 8 KW carrier and modulated 40 KW PEP. It would have done more if I used a vacuum tuner to replace the failing gamma matches.
Toll free is right, only way to get from 4 watts to the drive level required for a 3CX3000 in one stage is to go tetrode. Good luck finding a well built tetrode amp for 11 meters with shunt screen regulation. If you want to make it simple and use a triode, the 4 watt radio is going to need replacing with something like a FT-101, FT-757 series, or one of them "turbo" 100 watt CB's for AM.
Once you have 100 watts PEP drive from the radio on AM you can feed a single ceramic triode to drive the 3CX3000. The 3CX1000 would certainly run comfortably as a driver for the 3000 tube. Personally, if the goal was as few drive stages with low drive requirements, I would have built something like a 4CX5000 as a tetrode and hit it with a 100 watt radio. With all the MRI surplus tubes and Russian ones, combined with the "tetrode board" this is within the reach of a good amp builder.