If the tiles are attached, the Lauan can be removed w/the tile in place. Then, they can be taken outside and the Lauan Sheets can be 'broken up' a little to go in the trash. In short, the asbestos becomes a problem when the tiles generate dust. That can be done by using any type of saw (mechanical or manual), or by breaking them up in a SMALL 'space'. Doing the breaking up outside is kinda OK because you can situate yourself UPWIND when doing the breaking. Thus, you don't breathe in any of the dust. It's ALWAYS best to wear the appropriate Dust Mask (which is sold at the Home Centers.) Be sure to get the one for Asbestos Dust.
You shouldn't have to call in an Abatement Company UNLESS it's required in your Municipality. You can call your Municipal Inspector for 'guidance'. (Just hold off on giving him your name and address, if you know what I mean.)
I don't want to sound too dumb but you don't need/want to save the Lauan. Once it's used, you 'toss' it out. You're saying that you want to remove the tiles from the Lauan but forget that! Remove the ENTIRE setup and go from there.