No. You can't mix the native bar and BTT actions. This said, you can tell BTT wether you'd like to see the Default (macOS) bar or your own Preset depending on situations such as the program you're currently using, the document that's open, the window name,… You can add one and only one Control strip button that will appear and work in the macOS bar. Also, gestures (1 up to four finger gestures) can be set to work also on the native macOS touchbar.
On a personal note, I switched my touchbar to always show my BTT preset (check it out here).
But a two finger force click on the trackpad will switch between my preset and the Native Touch Bar. Super handsome. Also, if I hold ⌃⌥ the macOS bar will appear (totally functional of course) as long as these two keys are pressed. Check it out in your BTT settings!