Generally, .177 is the way to go. For all disciplines in FT.
If you are talking American Hunter Field Target (which is very different from what the Brits generally call HFT), the vast majority of shooters who do well shoot the 10.3gr JSB/FX/AA pellets. Some guys shoot the 13.4gr JSBs.
Since you are limited to 20FPE in the American discipline, you can push those 10.3s to about 920 which gives a VERY flat trajectory. This helps with longer shots where ranging error is more likely to be an issue.
I shot Open (also limited to 20 FPE) with a .22 AA S400 very briefly. You are definitely at a disadvantage shooting the .22. You can only push most pellets to ~750 or less before hitting the 20fpe limit. Trajectory becomes an issue here.
If you want to shoot .22, go for it. But there are a number of reasons why most folks shoot .177 for FT.