There are two currents for a relay, the carry current, and the breaking current.
The carry current is dictated by the thickness of wires, the contact area in good contact, and by the resulting \$I^2R\$ heating that occurs. This should be independent of AC or DC current flowing through the relay.
The breaking current is dictated by how fast the contacts extinguish an arc as they open, and as a result, how much damage that arc does to the contact surfaces. The factors here are contact opening speed, contact material melting point, contact heat capacity, and whether the input current is steady, or goes through zero many times per second (DC or AC respectively).
As your switch can carry 6A of AC, I see no reason why it shouldn't also be able to carry 6A of DC, provided that it's never called upon to control that current. If this switch works purely as an enable, and the switch off is done by some other switch, say a travel limit switch or control switch elsewhere, then you should be able to use it.
Although it might appear that switch on is far kinder than switch off, bear in mind that contact bounce means that switch on is usually followed by a switch off mS later, and then a few more bounces, before it settles on. If the current has built in that time to a damaging level, then contact damage will be done.