A small FPGA or CPLD and some fast external SRAM would do this easily, and less than $40 in part, but not once you've put it on a PCB.
Do you mean 10 bytes or 10 bits? 10 bytes is doable but you'd have a very wide bus, and will be well out of your price target
Many FPGA devboards have SRAM, but you'll have to pay a little more. You will probably struggle to find a board with fast enough SRAM, but slower, wider RAM is another option.
Probably the cheapest would be to use a reasonably wide SDRAM with a FIFO in the FPGA to cover the setup periods.
For PC download something with onboard USB, would be useful to make a single-board ready made solution - I'm sure there is something out there, but not for $40.
This product might be an option. There are plenty more out there.