Recently had to heat seal some 100% poly caps and was worried the heat and pressure from the hat press would "shine up" the fabric on the hat.
We solve this issue on shirts by reducing the pressure from the press and it works fine. This however is not an option on the hat press especially with structured caps as reduced pressure yields uneven contact between cap & vinyl graphic.
I kind of solved the problem by an alternate heat and press method using a sealing iron I use on my R/C airplanes.
The temp is plenty high enough from this iron and it is small enough to allow me to load the hat on the press and use the press pad as the support for the hat while hand pressing.
The iron can be bought for around $20.00 bucks.
link is below there are other sources and similar irons but this should get you going if you are interested.
mooseman
https://www.horizonhobby.com/product/sealing-iron/HAN101.html