The Precision Tower 5810 has a firmware upgrade available to bring the stock TPM 1.2 up to TPM 2.0, which will allow the machine to run Windows 11. It won't let you upgrade from an older version of Windows, it will still give you an "Unsupported CPU" error, but if you do a fresh install using USB/DVD media, you won't get a warning and the install will go fine.
With Windows 11 requiring TPM 2.0 I found myself in the same situation as OP. I did finally get it going but the process was incredibly frustrating, here we go: Update to the latest BIOS, Go into the BIOS\Security\ a new option will be there TPM1.2 (assuming you were on an out dated BIOS version like myself), Turn on AND enable TPM, boot into Windows and run the TPM 1.2 to 2.0 utility from support site (IF it errors out because it's "owned" you will have to boot back into BIOS and clear the TPM then boot back in), the PC will reboot, you will see the familiar "firmware update" screen, it might fail but try again like I had to do, once you get the Firmware update was successful message let me save you hours of frustration, boot back into the BIOS and see that the TPM still shows 1.2, navigate to "reset factory defaults", boot back into BIOS and finally you will see TPM2.0 and can enable it. Why on earth doesn't it change on its own, why isn't this just part of the BIOS Flash? I'm not sure but at least it can run 11 now, hope this helps someone