Use an eSATA to a SATA drive HDD caddy
Solution 1:
The only differences between Sata and eSata are the connector style and eSata cables are shielded. It should work. You may have problems when you connect it while booted if AHCI is not enabled in the bios and software is not installed for it.
AHCI allows for hot plugging of Sata.
If AHCI is not enabled in the bios or does not have the option to use it, that is a whole other problem to solve, you will have to connect and power up the external drive before you Boot the PC if AHCI is not enabled or not available.
Solution 2:
It's a little bit of a kludge, but I don't see any reason it shouldn't work fine. Some motherboards do have the eSATA ports on a different controller which can hurt performance and/or impair your ability to boot from the drive but PROBABLY it should work. I think it's a pretty clever use of the resources you have handy.