Your double pole switch will not be adequate, though it will work. The
requirement is for a switch that will prevent both line and generator to be
connected at the same time. The DPST switch will allow you to do that, but
it will also allow you to forget to do it.
Someone suggested a DPDT switch. That would work if there is an
intermediate position in which neither is connected, or else you have the
same problem. And, unless it is UL listed for that application it still
will not meet code.

You might want to confirm this by emailing the manufacturer, but I don't see
a problem with your generator on a 7500w transfer switch. They have circuit
breakers in them, like what is in your breaker box. Your breakers don't
trip everytime you turn a motor on, does it? So I don't see why the
transfer switch should trip in a comparable situation. But it is easy
enough to check with the manufacturer.

The generlink someone recommended looks like a pretty neat solution; except
installation might be expensive. Just don't forget that you still have only
7000w, even though every circuit in the house is hooked to it!