I'm guessing you want the shortest carabiner along the major axis? REI has the dimensions listed for most carabiners. Grivel Plume K3G and Grivel Plume Nut K3N are the shortest locking carabiners I can find to date at 3.5 inches. A triangle quick link (the one linked below is 2.75 inches) will be shorter but heavier than these biners. The shorter a carabiner is the smaller the gate opening is. This doesn't work well in the climbing community where every second counts trying to clip in. So they have to strike a balance of size when designing the carabiners. Also watch @Nutterbuster video on his tether and bridge in his saddle setup
I would recommend getting the wire gate. It's dummy (not saying anyone is a dummy) proof.
DanO sells it https://doublesteps.com/product/grivel-plume-twin-gate-carabiner/
I personally run Edelrid Pure Slider biners (not on bridge and also dummy proof). But they are about 4 inches tall. Hope this helps
Plume K3G
Compact and light carabiner with Twin-Gate system (wire-wire). This minimal carabiner is optimized for lightness and durability. Its ergonomic shape makes it easy to handle and operate. It features the innovative Twin-Gate system, wire-wire version, which does not require any additional maneuver...
grivel.com
Plume Nut K3N
The lightest and most compact screw-gate carabiner. Weighing just 37 grams, this super compact screw-gate carabiner is an extremely light device, capable of providing safety without compromising. Individually tested, with the date of testing and progressive number laser-marked on it. Activities...
grivel.com
Maillon Delta Plated Quick Link 8mm
Plated Steel quick link for use in 1-sticking and SRT for a girth hitch. Allows you to quickly attach your rope instead of feeding your whole line through your girth hitch. Better than a carabiner …
doublesteps.com
I would recommend getting the wire gate. It's dummy (not saying anyone is a dummy) proof.
DanO sells it https://doublesteps.com/product/grivel-plume-twin-gate-carabiner/
I personally run Edelrid Pure Slider biners (not on bridge and also dummy proof). But they are about 4 inches tall. Hope this helps