Well, use your gesture to show/hide the application - I use a keyboard shortcut as below - and bonus! the keyboard shortcut above it gives you easy access to another BTT fab feature... the built in clipboard manager (so pressing cmd-option-v shows the clipboard manager rather than the simple command-V to paste the last copied item only)
View attachment 1722977
psps I do have in Forklift preferences-general the option to 'restore windwos and tabs on startup' checked
plus forklift is launched hidden on login