I've used both the F1 Shooting Crony, and the Caldwell. I love the Caldwell, and it has operated flawlessly with both my iPhone, and iPad. I've heard that connectivity to other brands of smartphones can be iffy.
The Caldwell, without a phone, will only register the last shot. But the app on the phone is fantastic, and support (for me) has been excellent. I emailed in a bug report that apparently only existed with an older iPhone 4, and they released an update in less than 2 weeks that fixed them. I bought the deluxe setup that came with tripod (cheap, but functions for me), powered screens (which I only used once just to test) and carry bag.
The F1 Chrony will do shot group statistics in standalone mode, but I ended up getting the remote printer for it. The Chrony I liked, the printer was a bit of a disappointment. It works, but dealing with paper feed issues, especially on humid days, was a problem. If you get a Shooting Chrony, get the "Master" model, which gives you a remote interface. Easier to read and control without having to go forward of the firing line.
One optical chrono that I would like to see in operation is the newer G2 model from Caldwell.
https://www.midwayusa.com/product/169483/caldwell-ballistic-precision-chronograph-g2
The orientation of this chrono would alleviate much of the problem of lighting and sensors. It has built-in bluetooth connectivity. It is also inside your budget. I have not seen one in use, though, so I can't comment on that.