Unless you can prove me otherwise I don't believe there is a commercially available kit that is 'fit for purpose'. The first aid 'kits' that you buy seem to be aimed at minor cuts/blisters and scrapes, the sort of thing that you wouldn't come to much harm if left totally untreated. Where a first aid kit matters are situations of life and death, with some comfort items if space allows.
My take would be to decide what are the most serious injuries that you could sustain and what is most likely. Another aspect is that a lot of shooting is a solitary activity and what can be used for self care or a single item that can take on a number of separate uses. A serious laceration is probably the most severe life threataning injury you may sustain and a large pressure dressing is needed here. If you may be treating other people then your personal protection is vital and some disposable gloves and a resuscitation face mask could go a long way to protect you.
This is my take on a kit for personal use for various outdoor activities. I have a very much more substantial kit in the car with enough stuff to deal with more serious injuries such as an RTA.
The bag is about 15cm by 8cm and 6cm deep. I replaced the zips with paracord and large ends to make it quiet and easy to open if injured and self-treating. It folds out and I usually have a pair of nitrile gloves free in the middle. It fits in a bumbag or wax jacket pocket, but does have a belt loop that I personally don't use.
The listing is as follows:
2 pairs nitrile gloves
1 size 4 vacuum packed trauma dressing (with elasticated pressure bandage)
2 hypodermic needles (splinters)
2 sterile scalpel blades
Roll of 25mm elastoplast
small pack of mixed plasters (good quality)
Lip balm
2 20ml saline eye wash pods
resuscitation face shield
antiseptic wipes
2 blood stopper nasal plugs
2 small sachets of burn gel
clinical waste bag
2 ibuprofen and codeine tablets (reasonably strong painkiller/anti-inflammatory)
6 loperamide (generic Immodium)
4 chlorphenamine (anti-histamine)
My biggest dilemma is wanting to add ever more stuff to it! I have a draw full of spare first aid gear. In most of my coats I have a small kit with some gloves and a dressing pad and resuscitation shield. I carry a few plasters in my wallet. I have toyed with a haemostatic agent but realistically all bleeding will stop with sufficient pressure, we amputate cows claws and a pressure bandage is all that keeps them from bleeding to death. Also remember that if you deem a gun shot wound to be a significant risk you need 2 decent dressings (1 for entry 1 for exit wound).
The pressure dressings I use are these:
T4 Trauma Dressing Pad with Elasticated Bandage - SP Services (UK) Ltd
Buy a couple (even a training, non sterile version available) so you are familiar with their use. They even do them in cammo!