Does the m-die have an insert that can be used in the lee expander/flare die ?
NO.
At least not the dies I have. Other than the lock rings, its rare to find any part of one brand dies that fit and work on or in a different brand's dies. Sometimes you don't get part commonality even within the same brand, as designs get changed over time.
My M dies are Lyman All American, part of the 3 die pistol set, designed and made in those ancient dark days when the most "progressive" press was the Lyman Spar-T turret press. Back when Lyman presses were gray and red, not orange.
Dies came in a black fiberboard box, with a velvet tray inside. They are chromed (I assume, they are shiny compared to RCBS and Lee), and the set came with a die wrench. I have the .38 SPl/.357 and the .44SPL&Mag sets. Bought new in the 70s.
The expander die (M die) is a "Universal" open body (meaning no part of the die body touches the case), big enough to handle .45 caliber cases with a little room to spare. The expander plug shaft is just under a half inch diameter at the lower end, where the expander plug screws into to, and larger at the threaded top end (about .560") where it goes through the die body.
It is solid, not a "powder through" type. I checked with the Lee set I have (.32acp) and the Lyman M expander stem won't work, the shaft threads are too small to engage the Lee die body, by just a tiny amount. You might be able to make an adapter, but it would need to be very thin, and you'd still have a solid shaft, not a powder through type.
You could, with machine tools, take just the Lyman expander plug, and bore a hole through it, and possibly turn it to fit in the Lee die body, but it might be too short to activate the powder drop system. An additional spacer might be needed. Can't say for sure, you'd need to measure the actual parts you were going to use.
IF you want the Lyman M style "step" expander for a powder through die, I would suggest first going to Lyman and seeing what they currently offer.
If they don't have something that would work for you, then get a Lee powder through set up LARGER than what you need, and turn it down, to the desired diameters. Yes, two diameters, one for the inside of the case, and one for the "step".
I just did a quick and dirty check of the expander plug from my .44Mag set, and it has 3 diameters. .425" then the first step, .430" and then the "last step" which is the full diameter of the plug where it meets the shaft. .455"
You CAN flare the brass using the largest "step" but I don't recommend it.
I'm pretty sure just taking the guts out of a Lyman die and putting them in a Lee die won't work.
Good Luck!