The point and shoot cameras, even the expen$ive ones all have a shutter lag.
While its arguable whether the Sony RX100 mark IV fits into the OP's "won't break the bank" criteria, it most certainly meets it for speed. The "shutter lag" of the RX beats every Canon DSLR measured every which way according to the bench testing done here;
http://www.imaging-resource.com/PRODS/canon-7d-mark-ii/canon-7d-mark-iiA6.HTM and here;
http://www.imaging-resource.com/PRODS/sony-rx100-iv/sony-rx100-ivA6.HTM...and if you check out the Panasonic offerings, you'll see that the Sony RX is not alone for speed.
I have an RX-100 mark IV, and a later generation Canon DSLR (7D mark II), and I can tell you that the RX is the real deal for getting 'em off quick.
