I don't have a flyback. It seems to me that if you are up to changing one out, you may be able to arrange some other solutions.
There were a number of monitors that would sync down to Amiga frequencies (15kHz ?). I have seen them listed on the net.
The other computers you have connected to the 1084 are not using the analog RGB input. They use color composite input. There are many TVs with color composite input (stereo sound, too) available in Thrift shops since the HD standard has obsoleted them.
I understand Amiga video looks decent through S-video. If you can find a TV with that input you can be back in business, perhaps with more options for display.