I just got done trying a few things. Here's what I did.
I made a bin/cue file of my MechWarrior 2 disc, and saved it to my desktop. I then mounted the cue file in Alcohol 52% (which has a free version here), and ran the DOS install. For me, the virtual drive is drive F, and in DOSBox, I have it mounted as N (for reasons I explained earlier). So, I fired up DOSBox, changed the drive letter to N, and typed "install". This brings up the DOS v1.1 for Mech2 on my disc. I did a full install (very important), and then went to the page on Vogons that I linked to earlier. I grabbed the crack for the two executables from the link on that page, and pasted them over the original executable files. Once that was done, I unmounted the disc image, started up the game, and everything's peachy keen.
If you have the DOS v1.0 of Mech2, you need to upgrade it to v1.1. There's a link to the patch on that Vogons page that has the cracks (yes, the links all still work). Install the patch (I believe it must be run from within the Mech2 folder), and then paste the two cracked executables over the original ones. It has to be v1.1 of the DOS game for Mech2 to run in DOSBox.
Even if you have a cracked version, get those executables from the Vogos page, and paste them in anyway. This way, you're sure to have the executables that run fine in DOSBox.
So my suggestion is, delete the current Mech2 folder, to do a fresh full install of your game through DOSBox, make sure it's v1.1, and use those two cracked files from the Vogons page.