After some experimentation with my own K1000, I'm confident that it does not use the battery for the flash.
The camera's manual doesn't outright say this, but it does only mention the light meter when it talks about the battery. Wikipedia agrees.
The evidence that convinced me, though, was reading up on the X sync method, realizing that it is just a contact closure, and experimenting with it to find out how it works.
I reasoned that if the camera's internal battery were used for the flash trigger, I should see ~1.5 V between the two contacts, either continuously or momentarily when the shutter is closed. I tested this with a fast DMM and saw nothing.
Then I tested electrical conductivity between the hot shoe and the X terminal. The "ground" contact is connected all the time, and the "hot" contact is connected only when the two switches on each side of the hot shoe are depressed. (The square cross-sectioned sort of toothpick works great for this.) You can see these switches if you look under the little angle brackets that hold the flash on the shoe.
That lead me to try tripping the shutter with my DMM in continuity mode, with the probes across the hot shoe. With each shutter release, the contacts were closed momentarily. The same happened with the X sync connector. (You need a fairly fast DMM to see this. Cheap DMMs generally need more than 1/60 sec to sense a contact closure.)
I assume that this is driven by the mechanical shutter. Two contacts sliding past each other, touching momentarily to close the connection, kind of thing.
All a flash needs to do to sense this is to put a high-impedance voltage on the "hot" pin in parallel with a voltage sensor of some sort. When the contacts close, the voltage source will be dragged to 0 momentarily, which means the shutter just opened.
This is why the camera manual makes a point of telling you that the hot shoe contacts are cold when you use the X sync terminal. Without those switches you see in the shoe, the sense voltage put out by the strobe connected to the X terminal would appear in the hot shoe.
The manual talks about a risk of electrocution. One way I can see that happening is if you have a wall-powered strobe system that's malfunctioned, so that it puts out AC wall voltage to the X terminal. If that appeared on the hot shoe, you could indeed be electrocuted. Such a circumstance wouldn't necessarily damage the camera or blow a fuse.
The comment below tells of another way this could happen, which is probably more likely.
The reverse case — hot shoe to X terminal — isn't as well protected. The X sync contacts are easily bridged by a finger, and the plastic cap is easily lost. I assume this case isn't as worth protecting against since a battery-powered hot shoe flash shouldn't be sending its high-voltage flash power through the hot shoe, but instead a low-power sense voltage. I have not tested that, so just take it as a guess, not a statement of fact.