I fail to see how you can blow a fuse from your battery to converter, are you towing it plugged in to a 120VAC source?
My current TV does not supply full amps unless gain is set to max and brakes are applied hard. Old style aftermarket controllers were generally on or off, gain sets amps. Depending on your TV and how you apply brakes you will likely see different amp readings.
How are you checking amps for breakaway system?
Disconnect trailer from TV and 120VAC. Don't test while connected to anything, 12V backfed into your brake controller may not make it happy...
Connect amp meter between battery and breakaway switch.
Pull breakaway switch.
On each individual electric brake magnet the amperage draw would be 3.0 to 3.2 max amps at 12 to 13 volts on 7-inch brake magnets and 3.2 to 4.0 max amps at 12 to 13 volts on 10 and 12-inch brake magnets.
Per regs the breakaway system should be able to work for 15 min., don't leave it on.