Tj,
If the HO cathode current was within spec before the eff.coil went springy, very probably it'll be within spec with the new coil once you dip it using the 'magic light bulb' method.
Does the HO tube look beat up with heavy black deposits inside the glass or other obvious 'burnt' looks? If not, it was probably running in the safe range.
RS still sells #44 bulbs (cat.# 272-1108).
Kramden66 mentioned using a cap with the meter causing an open-circuit condition (in the cathode leg). He mighta had the cap in series instead of in parallel with the meter.
I never used a capacitor with the meter or saw any need to, since the meter presents a very low (effectively nil) impedance. Like a near-dead short.
Back in the day, there was an adapter (Pomona?) you'd plug into the HO tube socket, then plug the tube into the adapter; it breaks the cathode leg, and two leads coming out of the adapter connect to your milliameter. Doubt if those adapters are still available today, though.
(EDIT) Just a thought.. i never done it, but it
might be possible to use a
digital meter for cathode current by shunting it with a very large capacitor to smooth out the raggedy waveform. Several thousand mfd. might keep the meter from going nuts.