Your cardboard borders are the answer to why gentle: the edges are where the layers SHOW. Press hard and you weld the two sheets into one solid lip - no layers left to rise, hence the dense flat frame around a puffy middle.
The balance that works: a light fork press plus a water spray on the edge before sealing. The water does the holding (it glues the sheets), the fork just commits them - it doesn't need force. Sealed this way the rim still has living lamination and puffs WITH the centre.
If filling still escapes with a gentle seal, the problem is usually upstream: too much filling (less than you think is the rule), filling too wet (thick, cooled fillings only), or filling smeared onto the sealing border itself - keep a clean dry margin, fat and moisture on the seal line break any press, hard or soft.
Modest filling, dry margin, water spray, gentle fork. Jailbreak solved without killing the edges.
