There's not actually anything magical about DSLRs that give them low shutter lag. They have low shutter lag purely because they have fairly powerful computers in them for what they are.
I'm sure there are some high end point and shoots or if not, mirrorless cameras as well that have very low shutter lag. Not that I'm advising you against a DSLR, by no means. Just mentioning options. And lots of different kinds of cameras can have mechanical shutters for 1/4000-1/8000th second.
These are both usually advertised numbers, so it should be really easy to confirm whether or not a camera makes the cut just by glancing at its stats online.
There is some magic to the typical DSLR shutter, as it is an electro/mechanical shutter. This is usually a focal plane dual curtain shutter or it can be a leaf shutter. Most P&S cameras have no mechanical shutter, rather they use an electronic sensor that turns on and off the light reading ability of the sensor. The electronic shutter by it's very nature has a greater inherent lag time.
I said there's nothing magical about
DSLRs, not "DSLR shutters." As in, a camera being a single lens reflex camera by definition has nothing at all to do with its shutter lag or its shutter speed. These things are limited mainly by processor power and shutter actuation, respectively, not by the presence or absence of a mirror. If anything, the presence of a mirror, which has to be flipped out of the way, only serves as a potential
impediment to shutter lag compared to mirrorless designs.
Yes, a camera being a DSLR may happen to be correlated with high quality processors and shutters. But that has more to do with them being generally expensive and thus coming with options like this. It's not a necessary relationship or an exclusive relationship, and I just want to make that clear to the OP so that he does not misunderstand and restrict his search to only DSLRs. Other options may happen to be better fits for him, and can and do often have equally fast shutters and equally low lag (again, to verify this, both numbers are usually advertised in plain sight online, so there shouldn't be any surprises, and nothing has to be taken on faith).
For example, the Olympus OM-D mirrorless versus the Canon 6D full frame DSLR:
1) The mirrorless has a center single point AF shutter lag of 0.277 seconds, versus the more expensive DSLR's center single point AF shutter lag of 0.290.
2) Both cameras go to 1/4000th of a second.
3) The mirrorless shoots with double the FPS.