There are better glues to use than Liquid Nails for gluing the joint. I'd use either Tightbond III (waterproof) or Gorilla Glue. Then glue the 1/4" top and bottom as you planned. This will add a little weight over the original, but it will be strong. The Tightbond III will work better than the Gorilla Glue for this part. You can apply the glue for the lamination with a small 3" paint roller also from HD. Apply weight distributed on 2Xs to the layers of plywood for a good glue job.
My reason for recommending something besides Liquid Nails is that it doesn't flow out well for most glue jobs. It works well on floor joists when attaching subflooring with nails or screws. LN is applied from a tube with a caulking gun - it goes down in a bead, it doesn't flow out to smoothly cover the surface to be glued. Tightbond III when applied with a small roller makes sure there's glue completely covering the surfaces insuring a truly laminated solid piece of wood in the end.
Bill
Bill