The anti-siphon aspect of your sillcock is working as intended unfortunately. Any manufacturers of that type will recommend NOT installing hose timers that require constant pressure to be installed on them.
From one of the largest hose bibb/sillcock manufacturers:
We do not recommend using a hose timer if you have an anti-siphon hose bib or frost-proof wall hydrant. While great for irrigation purposes, Timers require you to leave your hose bib or frost-proof wall hydrant turned on with continuous pressure for long periods. Leaving your hose bib or frost-proof wall hydrant open for extended periods is against the ASSE 1019 certification for backflow prevention. The constant pressure, even if the water is not flowing, will cause damage to the backflow prevention check valve.
You could have a standard non-siphon sillcock installed, which technically would be against code in most areas, but could outfit it like this and still offer the protection of the vacuum breaker.