Why is my timer tripping the breaker on my pool filters power cable?
Switching GFCIs line-side tends to make them a bit cross

Plug-type GFCI devices revert to the "tripped" state when unpowered, unlike receptacle-type designs, which mechanically latch in both "tripped" and "untripped" states.  This was done to allow them to protect against an open neutral, which'd otherwise defeat the GFCI protection and make it impossible for it to trip.

However, this makes them incompatible with being switched on the line side, as you see here.  The solution is to move the GFCI protection to a location upstream of the timer -- GFCI breakers are ideal here, but a receptacle or deadfront GFCI can be used upstream of wherever the switch or timer is.  Once you've done this, you'll have to remove the GFCI plug from the end of the plug-in pool's cord and replace it with a regular plug, with a big note on it saying "FOR USE IN GFCI PROTECTED OUTLETS ONLY".

Either that, or you can remove the plug and wire it into a timer that's then wired in series with another length of appropriate cordage that is terminated in the existing GFCI plug, effectively moving the existing GFCI upstream of the timer...