There is no such thing as a class 3 weapons permit. In the USA small arms are either title 2 (machine guns, silencers, SBS, SBR, DD, AOW) or title 1 (everything else). Class 3 refers to an FFL that has paid the Special Occupational tax class 3 to deal in title 2 weapons. It even says this in the link you posted.
All I need to make or buy a silencer is the ATF form 1 or 4, 5220.20 fingerprints and photos and a $200 check. The ATF form 1/4 is not a permit or a license, it is just proof the tax was paid. It is easier but more time consuming than getting a first time driver's license.
Opps, I see divemedic has already addressed this.