People who make homebrew CPUs (have you looked at any?) tend to use SRAM for registers. Nobody in their right mind would solder up a load of flip-flops. Never mind affording it, the power needs, and the chances of getting it wired up properly.
You don't need dual-ported if you only ever do one read or write at a time. So to, eg, INC a register, have your CPU read it on one cycle, into a buffer. Increment the buffer in the next cycle, then write it back in a third. Time-multiplexing!
Some sort of buffering will be needed if you're feeding 2 registers' contents into a ALU. You could perhaps use just one buffer and get the second operand "live" from the SRAM. But of course there's no "increment" pin on an SRAM chip! You'll figure out where the buffering needs to be.
That said, there are 74-series registers. Originally entire CPUs were made of 74-series, or at least discrete logic chips, before the 74 series was invented. Searching "74 series register file" gave a few leads. Though of course just because it was made once doesn't mean you'll find it now.
Have you looked into FPGAs, or even CPLDs and PALs? PALs are too small to do a CPU with, but a few of them mixed in with the other logic might save you a few chips. In an FPGA though you could implement entire CPUs. FPGAs are basically thousands of logic gates on a chip. You can choose what logic each gate does, and how they are connected. You do this by writing code, like software. Then shoot the results down a USB lead to a programmer.
FPGAs are used a lot in consumer goods, and in many many other fields.