You may be presenting more arguments, but I think testing what you're suggesting is currently out of scope, considering it's a commander revamp patch and the only two (?) changes outside the revamped commanders were rather small and critical. It's mostly a community patch, with a balance team that doesn't have full authority over what they can and can't do.
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That said - IMO
I think such an artillery pit would simply be ignored by British players and UKF would just become the fourth faction with a mobile mortar. A static emplacement that gives 15% more range and only gives extra protection against small arms doesn't really seem worth the 100mp investment, especially when only getting one mortar. The strong, static mortar pit is part of what makes UKF unique, so if you want to maintain that uniqueness while giving both options, you'd have to atleast make the mobile option weaker than those of other factions, while making the static option stronger. This seems pretty hard to do right to me.
If you simply want the mortar pit to be more flexible, I think a dismantle option would be a step in the right direction. That way UKF has more control over their popcap and can react better to shifting engagements. The resources you get back could be made to scale with vet, starting at 33% (at 100% hp) up to 66% for example. This way, the Brit player will try to get the most out of it, while giving the axis player more incentive to kill it. Ofcourse, the dismantle option would take some time and can't be selected while being braced or in combat. What do you think about such a change?
I don't feel the need to present any other arguments, the mod is on the workshop and available for testing and I've done it, and like the OKW trucks being able to unsetup and move it feels organic, strange but something natural that has been there from the very start. Further more wherever you go and search for British mortar, be it here, the Steam forums or even reddit, you will see that the discussion ends with the same damn conclusion, a mobile mortar.
So while your idea of a withdraw and refit for it sounds good on paper it's still a gimmick, and as it's proven we need less gimmicks and more tried and tested solutions which are implemented everywhere else but for some reason, not for the British.
About the Artillery Pit specifically, I believe that it could be balanced so that the mortar teams would be cheap, say around 220-240 manpower but limited, similar to how the USF's mortar functioned before it was brought back into a normal mortar state. Difference here would be that garrisoning them into the Artillery Pit (which doesn't necessarily need to be 100 manpower exactly, could be more or less depending on the balance) would bring them to their full potential, maintaining the British's uniqueness while not sacrificing practical and viable mobile indirect fire.
So in short, the mortar team without being garrisoned could act as a mobile smoke dispenser which the British currently lack, with limited range and perhaps normal or a bit quicker tear down time, I am not so sure about the rate of fire however, I just know that the mortar's range was upped later in the war with the use of a new charge. While garrisoned into the Artillery Pit they would regain their normal range like other mortars and perhaps gain the ability to fire creeping smoke barrages and/or White Phosphorus like it's currently implemented, I think that that will give you a sense of a good investment into the Arty Pit plus the slight protection they will also provide to the mortars without making it feel like a waste.
In general if you want to remain mobile you sacrifice power but have mobility, if you sacrifice mobility you gain power, that's basically what the essence of the British is, especially with the Bofors, AEC and the Anvil and Hammer tactics, the system would just be a bit more flexible that it would not lock you out, that's all, but you would still need to make a choice instead of being forced into one.