I have a dog waterer that has a female garden hose connection. I would like to take a tee off my 3/8 compression fitting above my cut off valve under the sink to connect to this. Any suggestions on the best route to go? I would prefer to stick with a short braided hose like you would use for a toilet to make the run... But I have no idea how to take a compression fitting to a garden hose.
Not knowing what your valves under your sink look like , I would suggest possibly a new valve with 2 ports or the addition of a T on a 3/8 fitting then you could find a pre made hose like you want for a Refrigerator, this can feed your waterer.
I would want an independent shut off incase of problems, my plumbing speciality store could set up all the parts needed and suggest things like an extra valve just in case by using a photo of your existing under sink set up. They may cost a few dollars more but 1 trip ends up saving time and frustration in the long run compared to the big box stores , the big box may have what you need but getting the correct help sometimes it tough.
I know some may respond that this is not an answer to the question but sometimes it is necessary to shed light on why the question may be trying to solve the wrong thing.
I second the comment from @jsotola. The hose fitting connection should make it obvious that the dog watering unit is made for use outdoors. You do not want to hard connect the watering unit indoors because a feisty dog could easily overturn the unit and you may return one day to a flooded house.
If you need an indoor unit get one of the types that recirculates a limited supply of water to keep it fresh and then refill it as needed. In the case of these the water spillage in an overturn is much much less.