Good news: you can use this generator with this transfer switch
The good news is that the EU7000is, luckily, is a floating neutral generator, so it can be used with simpler transfer switches, such as your "select circuit" switch, or breaker interlocks without violating the NEC's grounding and bonding provisions. (Many jobsite generators have bonded neutrals as per the OSHA rules on worksite use, but this means that they need to be used as separately derived systems. This, in turn, requires a transfer switch which switches the neutral wire over so that you don't put the two neutral-ground bonds in parallel inadvertently, which'd cause merry havoc with stray neutral currents and even GFCI tripping.)
...but the cord you'll need is kind of odd
The problem is that you'll basically need an adapter in order to plug your generator into your house, as your transfer switch has a 50A, California Standard twistlock inlet which requires the correct connector (the inlet's a CS6375 and the mating cord cap's a CS6364), while your generator has a NEMA L14-30R on it. Such a thing does exist, or you could use a length of 8/4 outdoor cordage (such as SOOW), a NEMA L14-30P, and a CS6364 cordcap to fabricate your own cord.