Shooting a reproduction shot as tiles is absolutely normal, and standard operating procedure. It's just not done with the sort of makeshift and low-budget set-up you've been describing to us over a number of questions very often, at least not on a regular/professional/production basis. (And it was a regular thing before digital photography as well.)

Ideally, you would want a setup very similar to a rostrum camera. Either the camera or the painting you're shooting can be the moving element - normally, the thing to be copied is on a table that can be moved in X and Y directions if it's relatively small, and for larger or more fragile things the camera is moved (using a gantry on tracks for a horizontal set-up, or a vertically-adjustable stand on tracks for a vertical set-up). Using a tripod, things get a bit trickier, especially if you're not free to build a jury-rigged track-and-dolly system. Basically, you're being told to do the right thing, but not being allowed to use any tools that would allow you to do the right thing. Which is a pity, since the cost of making an adequate shooting setup - the materials cost, at least - would be about the same as the utterly inadequate tripod solution (which is almost never the right choice for working in a single location/studio). No, you wouldn't be able to take it outside or on location, but you'd be able to process a lot more items a lot more quickly.