Your overall issue is going to be that acrylic is a thermoplastic and gets soft when it gets warm, and as you’ve identified, you’re throwing heat into what is ultimately a very thin piece of plastic. I’d say you have three possibilities for how to handle it.
Option one, minimize the heat.
Good idea on the cutting alternating lines to keep the heat buildup to a minimum. What if you took that to the extreme; broke each line up into segments and cut a short segment? or cut your lines, but leave a bar along the bottom connecting all the fingers, then your last pass is a horizontal cut that removes the bar? Maybe even try pre-chilling the acrylic just before you drop it in the machine. I wonder how it would behave coming out of the freezer? Though looking at the width of your individual finger pieces, I’m guessing that they wouldn’t have much thermal mass.
Option two, change the material.
Perhaps try with a material that doesn’t soften at relatively low temperatures. It looks like you’re making gel electrophoresis combs, so I’m guessing wood is out. However, maybe look around for a material that would be compatible with your needs but is a thermoset vs a thermoplastic. Thermoset material, once formed doesn’t melt, meaning that the residual heat left by the laser wouldn’t be softening the plastic. That said BE CAREFUL ON YOUR CHOICE OF PLASTIC! Do some research because cutting some plastics can release chlorine or cyanide gas that could kill you, or worse, destroy your laser.
Option 3, reshape your piece after cutting.
This is probably what I would try first. The idea is that you’re already getting the right topography of the piece, you just have to reshape it. I’d cut out a frame that’s the inverse of the piece you want out of something like 1/8in plywood and use it as a template for your final product. Heat the acrylic gently, maybe with a heat gun, or by leaving it in the oven on warm till it’s soft, and then lay it into the wooden frame to hold it into the proper shape till cool.