The easiest and cheapest way is to get a Genlock....preferably with S-Vdeo OUT and not just composite OUT...then you can connect it to a S-Video TV..the output is very close to that of an RGB monitor.
Get a SuperGEN SX genlock if you can.
Then you can watch TV and the Amiga at he same tim throuh just moving the Faders up and down to switch.
Or........Just trade me your A4000 for my A3000 with Monitor included....the A3000 has VGA out with FlickerFixer built in! as well as a sepereate RGB OUT. :-D