Check the picture tube filament with an ohmmeter first. As you say, if the tube's a dud, you may not want to invest a lot of money and time in the TV until you have located a replacement.
It's premature to buy a tube tester. Many vintage testers will need some repair, just like the vintage TV you're proposing to fix. You don't want to be one of those guys who ends up with a garage full of broken stuff and no idea where to go next.
You can do the same filament test on the TV's small tubes. If they pass this basic dead-or-alive test, many of them (perhaps all of them) will work OK in the TV.
You could also contact an area collector club and ask whether someone there has a tester that you can borrow. Or, perhaps you could bring your tubes there for testing. Here's a list of clubs:
http://www.antiqueradio.com/clublist.html
Phil Nelson