The Bleeding card states

“During your activation, after you resolve an action other than the action listed below, you suffer 1 .

: Discard this condition.”

As Fenn’s Tactical Movement is an ability and not an action, can he use TM without suffering 1 ?

Yes. Things that are not actions do not trigger the Bleeding effect.

Tactical Movement is not an action, so it does not trigger bleeding.

A hero suffering strain to gain movement point up to twice per activation is not an action, so it doesn't trigger bleeding either.

Spending movement points is not an action, so it does not trigger bleeding.

A hero cannot voluntarily suffer strain over his endurance, but bleeding is not voluntarily suffering strain. So, order matters when you want to be clever and not remove bleeding.

Also note that a hero can spend a surge while attacking to recover 1 strain (recovers damage when the hero has not suffered strain). In an attack action this would first recover strain, and after the attack suffer strain from bleeding.