If your fluke is a class 3 then it should be fine, assuming its a 'fluke', that name and 'avo' seem to be used like the term 'hoover' does and apply to any old meter.
The fb doesnt usually cause a bright picture, and if the cathodes and grids are about the right voltage which you said they are earlier up, unless theres a bad connection to the picture tube base then maybe the tube has an internal short, especially if the machines been upended or led down.
I remember the days of the 'bulb bopper', those things raised a few eyelids, for dealing with a short however sometimes a cap charged to ht can improve things, but its fairly involved and a bit risky.