Generating pseudo-random numbers with restricted hardware
Is there any other clever solution?


If you are limited to 32 registers then you can only have 32 bits of state in your random numbers.  Therefore the only thing you can do to expand it out is use some combinational logic to produce the 448 bits that you need.

Generating the 32 bits is (as you surmise) probably most efficiently done using an LFSR.

I would suggest that using some kind of hash function of those bits would be your best bet to getting the least pattern passing through to the 448 bits output, but that might take up too much logic in your destination.  It's not clear to me what constraints you have there.