Gluing HDPE is nigh impossible! Please consider to create a sketch or two to better present the construction of your project. This will make things easier to provide useful advice. Quarter inch thick sheets will hold fasteners better than the thin stuff, so you're on a good track there.
One can use T-nuts with appropriately sized holes drilled in the sheet. The barbs on the nuts will secure the nut to the sheet, especially if a small diameter hole is drilled to provide a destination for the barb. If you use a T-nut, drill the pilot hole, place the nut, then tap it with a hammer to make marks for the small holes. The small holes should be 1/16" to 3/32" diameter, as you want the barb to "push" into the hole with some force, not simply float in a too-large opening.
From Amazon:
![3/8" T-nuts](https://cdn.statically.io/img/i.sstatic.net/1Uc95.jpg)
The link and the nuts shown are for 3/8" but one can find the 1/4" size easily enough.
Less secure for fastening is to use rivnuts, but especially with HDPE, they will probably spin in the hole and make fastening or removing a troublesome matter.
I've threaded UHMWPE of sufficient thickness with success and it cuts under a tap quite easily. It can also be easily stripped, which should be taken into consideration.