I honestly don’t like the shake awake feature either. It’s ok if you plan to leave the gun stationary in a safe, but in my experience if you’re planning to move the gun around the battery life will be significantly lower than the published runtime. I have a Sig Romeo 5 which is a decent dot that is “rated” to 50,000+ hours of battery life, but that is taking into account the power saving of not having the dot actually projecting at all times. I put a new battery in the optic and took it to a carbine class (hours of driving each way, so it was constantly on plus time actually using the gun during the class) and then returned to my normal usage schedule and the battery was dead after 2 months.
Unfortunately in order to get an optic that is actually constant on (as opposed to using a shake awake system) and still has the multi-year battery life you’ll probably want to move to an Aimpoint. It’s a bit above your stated budget, but I’d recommend either an Aimpoint Pro or an Aimpoint Duty.
Note that if you go with the Aimpoint Duty, it has the footprint of a micro, but it sits a little bit higher on the mount than a T2 will. T-Rex Arms has some pretty good pictures showing the comparison.