I guess a test that would tell if the eye tube was working properly or not would be to test it in a dark room with your eyes adjusted to the dark. If it slightly hurts your eyes at first or at least you can feel a slight strain even if there is no pain, then it's good. This all depends on how sensitive your eyes are, of course. You should be able to see it very well in a normally lighted room, and with your hand about a half foot away (perhaps a full foot), you should see a good green reflection on your hand even when near (not right next to) a 100 watt lamp.
Thomas
You can obtain wonderful results with 110 volts at the target with a new eye tube if properly selected, but changing the wiring as you stated above is probably a better idea for more flexibility. Be sure that your electrolytics are fresh for optimum B current. As for running your radio on a battery, if you live in the U.S.A., 6 volt battery chargers are fairly easy to obtain (you may already own one). They are usually sold as combination 6/12 volt chargers. A charger of 6 ampere capacity or greater is ideal, though no more than 10 amperes. You may try operating the radio from the battery with the charger connected. The battery will act like a stabilizer, but the charger will keep the system at full operation. The only problem with this is that your radio uses a low voltage 1.5 volt filamentary tube in the output section. The battery charger will likely boost the available voltage to the radio up to 6.5 or 6.8 volts with the battery connected. It can go even higher without the battery, though doubtful with the load of the vibrator supply. As long as the battery is put in place as a stabilizer, the voltage is not likely to go above 6.8 volts. This higher voltage is not a problem to either the indirectly heated 6 volt tubes or the vibrator supply. You must be sure that the voltage across the output tube's filament does not exceed 1.7 volts, though, or it will burn out prematurely. Ideally it should be at 1.5 volts. A small resistor may cure this. The other problem that may be had is hum with a filamentary tube, as the charger is not filtered. The battery will likely filter out this hum, though.
Also, I do not know where you purchase your 6 volt batteries from or what type you use, but if you have a Batteries Plus in your area, they sell 600 cold cranking ampere auto batteries for only about $45. I use one in my car all the time (1951 Chevrolet). It usually lasts 4 years with regular use, if not longer, and is capable of starting the car even when the temperature drops to 10 below zero. Such a fine but inexpensive battery will likely provide service to your radio for much longer if maintained properly. Just be careful with the acid problem and with the hydrogen off gassing while charging (explosive).
Thomas