Suggestions for 3-4 nights in Southwestern Germany

Maybe you would enjoy some time in the Black Forest. Gutach im Schwarzwald is a fun family-oriented destination in a scenic, well-located spot on the Black Forest Railway. The excellent open-air museum and summer bobsled are next door to one another:

https://www.tripadvisor.com/Attractions-g1825958-Activities-Gutach_im_Schwarzwald_Baden_Wurttemberg.html

The Black Forest Railway runs between Offenburg and Singen in the direction of Lake Constance (on the Swiss-German border.) Gutach has a recently-built station - not shown on this map, but it's between Haslach and Hausach.

https://upload.wikimedia.org/wikipedia/commons/4/4d/Verlaufskarte_Schwarzwaldbahn_%28Baden%29.png

Haslach is a pretty town with a Black Forest Costume museum, and an old silver mine you can visit (Segen Gottes.) Other nice places on the BF Railway and elsewhere are worth seeing as well - they include adorable Gengenbach (old walled town with towers, half-timbered buildings and a popular travel base town,) Schiltach, Donaueschingen (source of the Danube River,) Triberg, and Villingen-Schwenningen.

Schiltach

Small-town accommodations are available all around the Black Forest. This official tourist site will give you the most options:

https://www.blackforest-tourism.com/accommodation

We had a 2-BR apartment at this farmhouse in Steinach a few years ago. It was an easy walk to the station in Steinach for outings along the BF Railway. Like Gutach, Steinach is just one station from Haslach. Mr. Schoener was a great host.

Niedereschach is very close to Villingen-Schwenningen (also on the BF Railway.) (Notice there are lots of -ach towns. "Niedereschasch" is one of those two - "Niedereschach" is the correct spelling.) If you need a car during your stay for the wedding day, you can pick one up in Offenburg or in Villingen-Schwenningen. I would just take the train to V-S from your base town and pick it up there. It really should be no problem to use the trains from FRA airport to the Black Forest, around the Black Forest (for free,) and onward to Switzerland, if that's what you wish to do.

You don't want to take a car with you to Switzerland if you intend to drop it there - that will mean a hefty foreign drop-off charge. A pre-purchased train ticket from wherever you are in Germany to Zurich from the German Railways site will probably get you to Zurich inexpensively.

I forgot to mention that as guests in the Black Forest you will have free transportation on all the BF train lines and buses; make one of the participating towns your base to receive the KONUS card at no charge:

https://www.blackforest-tourism.com/info/KONUS

I loathe the idea of renting a car but think it may be unavoidable
given the location of the wedding.

Where exactly is the wedding? If it is in the town there is no need to the car, Niedereschach is just a short bus ride from the closest railway station.

The train planner of the Deutsche Bahn includes bus connections.
Niedereschach is less in the middle of nowhere than you seem to imagine. Donaueschingen, for example, is only 45 mins away.

My family lives in vicinity Niedereschach/Villingen and in Konstanz. July is the best possible time to visit Black Forest and Lake Constance. The kids would enjoy the festive lake towns, home-basing at Konstanz (1h10min hourly train to Villingen + 24 min to Niedereschach on bus 38/39 https://m.v-s-b.de/de/fahrplaene/ortschaften/43/).

If you fly into Frankfurt Airport, take the ICE train from the airport to Karlsruhe (1hr), and connect to the Karlsruhe-Konstanz hourly local train which goes thru the most scenic part of Black Forest (b/w Hausach to Donauechingen), stopping at the elegant spa town Baden-Baden, beautiful Gengenbach, Triberg water fall, quaint Donauechingen with the Fürstenberg palace and the fountain on palace ground reputed to be the source of Danube river.

The kids might enjoy Black Forest Open Air Museum in Gutach (also on local train line)
https://www.blackforest-tourism.com/Media/Attraktionen/Black-Forest-Open-Air-Museum-Vogtsbauernhof

Up to 5 people can travel with Baden-Wüttemberg ticket and visit as many towns as you want by local train in 1 day. https://www.bahn.de/regional/view/regionen/bawue/spezial2/swb_fahrplan_strecke.shtml. https://www.bahn.com/en/view/offers/regional/regional-day-tickets.shtml.

If you take the Karlsruhe-Knostanz train from Karlsruhe on your arrival day with luggage, there are lockers at Baden-Baden train station, Gengenbach has no locker but the tourist office will gladly keep them while you visit the town (the staff told me so when I stopped by Gengenbach for lunch 2 weeks ago).

Konstanz is a lively/friendly/festive university town with lots of beautiful old buildings, wonderful lakeside promenade, ferry going to cute Meersburg and Friedrichshafen (the interesting Zeppelin museum), and the beautiful island of Minau with its world famous garden/grounds. Since Konstanz is right next to Switzerland, a direct local train runs b/w Konstanz and Zurich in 1 hr 15min.

I loathe the idea of renting a car but think it may be unavoidable
given the location of the wedding.

I assume you decided renting a car would be unavoidable without looking up connections on the Bahn schedule website.

As pointed out above, there is a 25 min bus connection from Villingen Bahnhof to Niedereschasch (Rathaus).

The Schwarzwald Tourism website shows several vacation apartment with ¼ mi of the Rathaus bus-stop in Niederereschasch, as well as half a dozen in nearby Niederereschasch-Kappel, around another stop for that bus.

It might be a good idea to ask the au pair about where might be a good place to stay in Niederereschasch and what might be good transportation options to a car.

As far as airports, I believe Delta still has non-stop flights from Atlanta to Stuttgart. Stuttgart has train connections to Villingen Bahnhof.

We’ve stayed for several weeks on Lake Constance (Bodensee) twice and think families with kids would love it. Our favorite towns, with train connections to Zurich and the airport, are on the eastern shore of the lake.

We loved the warm water swimming areas, ferries across the lake, medieval walls and lively town centers, bicycle paths around the lake, promenade dining, etc.

Wow - this is all so much helpful information.

Also, regarding train/bus options I should have know to look beyond google maps:)

It's not necessary to go through Stuttgart Hbf. The Stuttgart airport is south of Stuttgart and so is Villingen Bhf. Take the S-bahn towards the Hbf only as far as Stuttgart-Rohr, then connect with an outgoing S-Bahn south to Böblingen, where you pick up the RE towards Villingen-Schwenningen.

Konstanz and Meersburg already have been recommended - and rightfully so!
Triberg (a famous Black Forest town) is pretty close to your location as well.