If you can't find an original bandswitch, pick the closest model Ameritron and order its bandswitch. It will have more contacts on it than you need for input circuits and 160 meters but you can get one that has much higher RF current ratings.
Circulating currents in the tank coil are greatest on the highest frequency band. Compound that with 11 meter users sharing the 10 meter tank coil and you have heating problems. I'd change the 10 meter coil to some copper tubing like the smallest refrigerator water line stuff.
Increasing the diameter of the conductor used to form the tank coil will require a longer length of conductor to reach the same value of inductance. Usually adding approximately 1 turn on the 10 meter coil will allow you to bend the coil to reach the match.
The MFJ is your friend here. Tune the amp for maximum output on any band you can get it to work at. Once you got it tuned for full wood, power everything down. Remove the tube from the socket and replace it with a 5000 ohm carbon potentiometer from the plate lead to ground.
Keep the wires from the plate clip and the chassis as short as possible when connecting them to the 5K pot. Connect the MFJ to the output of the amp set it to the frequency the amp was just tuned on. With no power to the amp, manually engage the RF relay and adjust the pot for minimum VSWR.
With the MFJ off, you can now measure across the 5K pot with the DVM and find the resonant load impedance of this amp. Any changes you make to the tank circuit should always allow a match back to this impedance. Now you upgrade the 10 meter coil and check it for a match without AC, RF or DC applied!
Just keep the pot in place and adjust the 10 meter coil, tune and load controls for a perfect VSWR on the MFJ. The coil should be bent so the plate tune adjusts close to its normal spot for the band in use.
If this amp is going to be driven with an HF rig, you can skip the neutralization. The control grid is already swamped to ground through a 50 ohm 100 watt resistor. The tetrode has now lost most of its gain over a triode. It should run stable without neutralization although IMD would be improved if it were added.
The biggest problem with adding it is the amp uses no input circuits. Without them you have no means to invert the neutralization signal and no ability to provide the correct level of feedback across the amps operating bandwidth.
I'd skip the neutralization in this case and favor adding some regulation to that poor screen supply. Especially if it's going on 80 meter SSB. That alone will make a big improvement in IMD and linearity. The zener diode across the base collector junction of an NPN transistor is a good way to make a heavy duty zener with improved regulation.
You may also notice this amp uses a Pi-L tank circuit. That helps reduce harmonics and is a real good thing on the lower HF bands where the harmonics would land in an upper ham band. The "L" section usually does not have RF heating problems. When choosing a bandswitch, make sure you pick one that can accommodate this circuit.