Rather than fiddle with 2mm gaps why not fill the void behing the boards and studds with inert permable insulation (rock wool). This would give an extra 50mm insulation and cover any cool spots that may have been caused by 2mm gaps.
If too late to fill the void I would put up a plastic wall or ceiling (not ground) vapour barrier, some may this overkill with foil backed plaster board and tape, but it is cheap and easy (untill you cut holes into it for switch & socket boxes).
If you are using a foil finished insulation board , don't bother with foil backed plaster board use standard beveled edged plaster board which is much cheaper.

I'm now on my third new build house using PU insulation in the timber framed walls (140mm!), it's not possible to make every insulation panel a perfect fit between the timber studs so my solution has always been to apply a bead of builders silcone round each panel to prevent any air movement bypassing the insulation; local building inspector reckons it's a good solution.

Personally I reckon vapour barriers in a PU insulated house to be a complete waste of time, even with tissue faced boards there is no danger of interstitial condensation. However you'd be hard pressed to find a structural engineer or building inspector who will categorically say that you don't need a VB, it's more of a "just put one in to be safe"; and as you quite rightly say, the first piercing of the wall makes it redundant!!

I honestly don't think you can have too much insulation, although there is a point at which extra insulation does not provide sufficient return on investment. A very well insulated property is more comfortable than a poorly insulated one no matter what the external conditions.

I find that with 140mm of PU in the walls, the same in the floor, 100mm in the loft space with 200mm of glass wool on top, then there is virtually no need for heating; just living in the house with appliances, lights, etc on is enough to keep it comfortable in all but the coldest weather. In addition it doesn't seem to overheat in warm weather in the way that rooms with uninsulated brick walls can do when the sun hits them and the heat is conducted through to the inside. You do have to make sure that if all parts of the property are not as well insulated you have sufficient control over the heating so that you only heat the parts that need it (a building regs requirement on new builds anyway).

You will no doubt be putting more heat into the room from catering equipment than would normally be the case in a domestic environment, that should simply mean that you won't need additional heating and if you do have to open windows occasionally you're only dumping heat that is a by-product of your business, but during cold weather it's great knowing that you're comfortably warm without spending extra money on extra heating

The problem with mega-insulation is the parts that remain uninsulated, particularly prone in retro-fit, which attract condensation because the effect of the general insulation is to make the masonry much much colder. Window reveals are a typical. (Incidentally, depending on the size of windows and wall specs, you may find that windows are a more deserving priority).
Panelled reveals (are they in fact really shutters?) may already be providing quite good insulation since wood is a good insulator and there will be an air gap behind. Any non-panelled window or external door reveals may be prone to condensation in severe circumstances. I would wait and see - it sounds like this is a cafe with potential for a lot of cooking steam - in which case a dehumidifier may be a valuable weapon. These incidentally return more heat than the electricity they consume because they also liberate the latent heat from the condensed water, which turns out to be quite significant in some cases.

As to insulation gaps in your studwork, why not use an expanding foam? This can be applied very precisely with a gun(Soudall gun recommended) - much better than the simple aerosol can. It also deals with all the pockets and service holes very effectively.

Mook: The most cost-effective way of prioritising funds is obviously to tackle aspects that give the greatest return for money spent. Usually the priorities are ceilings and walls, because there cheap to fix, but there are diminishing returns for the increasing thickness of insulation/complexity and a point is eventually reached where you will get a better return tackling eg windows than spending even more on the walls. A cheap DIY window solution might be secondary glaxing - which can be removed in summer. Draughts are a complex subject, because you need a certain rate of air change for health and comfort - and anyway social abuse (eg leaving doors open) undermines just about everything you do.