My thought is that we should wait until it has gone dormant for the winter, cut the branches back somewhat, and then dig it up, hoping that we've got a decent root mass to above-ground mass ratio.
I believe you are on the right track, thought process. Below is how I would go about moving/transplanting a mature gooseberry bush...
First, if you can wait until late Winter or very early Spring to dig it up and move it, that would be ideal.
Second, before digging it up, spend sometime choosing an appropriate location in your landscape and preparing that location.
Below is from your states University Extension Office, Fruits & Nuts: Landscaping with Fruit and Nut Crops (direct link to PDF)
Gooseberries grow normally in cooler climates, so heavy mulching of the base of these plants is desirable to keep roots cool. Some shade is permissible, but heavy shade is detrimental to growth and fruiting.
Seeing as your Summer's can be pretty (brutally) hot, I would be inclined to find a location that gets full morning sun, then offers some shade from 2 or 3 o'clock onwards.
If the soil in your chosen location isn't organically rich and/or has poor moisture retention, take the time to dig-in plenty of organic matter (eg compost, manure, shredded fall leaves). Taking the time to do this, if needed, will pay huge dividends once you come to transplant time.
Me, being me, I would use compost as the mulch layer. Using compost as a mulch has the added benefit of feeding plants naturally and slowly. Also if your soil is "poor" using a compost mulch layer will help improve the soil over time (as you add a "fresh" layer, once to twice a year).
- If you don't have access to homemade compost, locate a local resource and ensure you can have some on hand (enough for a 2 to 3inch / 50 to 75mm thick mulch layer) on the day you transplant.
Below come via Gooseberries and Currants
Site Selection and Soil Preparation
Unlike most other fruit crops, currants and gooseberries tolerate partial shade and prefer a cool, moist growing area. Northern slopes with protection from direct sun are ideal. Planting along the side of a building or shady arbor is suitable as well.
Avoid sites with poor air circulation, which increases the incidence of powdery mildew. Sloping ground alleviates this condition. Also avoid light-textured, sandy soils. Rich, well drained soils that have a high moisture holding capacity are best. Incorporate organic matter (compost, peat, or manure) to improve the soil, particularly if it is somewhat sandy. The ideal soil pH is about 6.5.
Transplant day (assuming later Winter or very early Spring)
- First, prune the gooseberry bush. Personally I would prune as heavily as I dare, make the physical size of the bush as small as possible, thus making manhandling of the bush as easy as possible...
Gooseberries produce fruit on 2-year-old canes, 3-year-old canes and older canes. New canes originate from the underground crown each season on vigorous plants. The younger canes are more productive, so it is desirable to remove the older, weaker canes by cutting them out at the soil line. This leaves a preponderance of the younger, more vigorous canes and encourages a continuous replacement of new canes. Young plants, therefore, require little pruning for the first three or four years.
Pruning - Currants and Gooseberries in the Home Garden
You should prune established currant and gooseberry shrubs to encourage vigor and fruit production, improve sun penetration into the bush, and maintain good air circulation to minimize disease. During the first three years of growth, allow four or five canes to develop per year, removing only weak or damaged wood. Beginning in the fourth year, prune out the oldest wood annually in early spring before growth begins. In addition, remove any weak new growth. A mature bush should have 9 to 12 canes once pruning is completed. Fruit is produced on one-, two-, and three-year-old wood.
After you've finished pruning, if the ground isn't wet, soak the soil around the bush.
Then dig all the way around the bush, dig as far out from the it as you can, you want to limit the damage as much as possible to the root system (which is shallow).
Work a tarp under and around the root-ball, this will help keep the root-ball encased in soil during the lifting and moving processes. Then when moving the bush, pick it up (manhandle it) via the tarp.
- If you haven't got a tarp, pick one up beforehand and have it ready to go.
Dig the new hole only as deep as you have to ie When you put the bush in its new location it shouldn't seat deeper in the soil than it did previously. Also dig the new hole twice as wide as the root-ball you managed to remove from the ground.
Back fill the hole with the material you dug out (the well prepared soil you had prepared months, weeks beforehand), making sure you heel it in well as you go.
Once the bush is in its new home in the ground, give it a long, slow drink, you really want to water it in well.
After that, put down your (compost) mulch layer.
Some of the above points come from this answer, Any special steps which should be taken when transplanting a potato bush? here on SE.
Below are some resources I used for this answer, plus additional information that I believe should prove helpful/useful:
Good luck! and I hope the above helps somewhat...