[Andy S]

"my guitar is only a cheapo les paul copy"

Where are they breaking? Near or on the bridge? Near the nut (by the tuning keys?)
Reason I ask, I've had two similar situations, both with different causes. I currently have a Dean Evo that is their version of a LP. I had made some adjustments and lowered the stop bar (the part that holds the end of the strings. I started breaking strings just after tuning & playing for a short while. In that situation, The string was rubbing up against the hole where the string comes out & up toward the nut. It rubbed enough so that after just a short bit of playing, "PING!" right at the point where it made contact with the stop bar. Solution was to raise the stop bar a bit. This changed the angle the strings were coming through the hole, no contact, no broken strings.

The other situation was a friends guitar. He changed string size (went smaller) and all of a sudden saw his strings breaking at the bridge itself when he brought them up to pitch and started to play. He went from .10 strings to .08. After looking at & feeling the saddle where the strings sat, we noticed a slight defect in the saddle. Enough to where the smaller strings (.08s)got damaged almost immediately but the original larger strings (.10) seemed less affected by it. I took some emery cloth, folded, and ran the edge through the saddle to remove the burr (defect, whatever it was). Problem gone. Similar to what UGB mentioned.

I do agree with UGB, though. It sounds , from your description, that they are breaking as you try to tune it and that it may be an issue with the tuner not responding fast enough and you overshooting the target, then going to the next octave.

Keep us posted on your progress. Good Luck!