Konigsee with a toddler

Sorry to say, but you can cross the Jennerbahn off of your list as it is being upgraded this year thru Spring 2018: http://www.jennerbahn.de/de/home/
To the Oberseee with a stroller doable but not to the Fischunkelalm (unless you are ready for a challenge and carrying a lot both toddler and stroller) as there are some steps and narrow rocky paths. No way to walk around the Oberseee unless you climb or swim. http://www.fotocommunity.de/photo/fischunkelalm-hartl-johann/24654152 Plenty of places to walk with an unpaved capable stroller at St. Bartholomew. The beauty of the Königssee is worth the visit with good weather. I keep going back. Nice Beer Garden at Gaststätte St. Bartholomä: http://www.bartholomae-wirt.de/

The boat ride on the Konigsee is definitely worth doing. If you have a small umbrella-type stroller that folds up and down quickly I think this is doable. (In fact, I would do all of your touring about with the smallest stroller possible. ) That is a tricky age because the "little" may not want to sit in a stroller all the time and prefer to be out and moving. That would be fine for most of those locations. There is a lot of equipment available that would allow you to carry your toddler on your back for locations where the stroller doesn't work, so that might be another compromise that would allow all of you to enjoy your time.

When we traveled with our 22 month toddler for a two week trip, we used a Osprey Poco AG Plus Child Carrier from REI. https://www.rei.com/product/895652/osprey-poco-ag-plus-child-carrier

It is very adjustable and very comfortable on long days with her on my or my wife's back. You strap her in, is has a sun shade. We attached several toys and a pacifier to straps attached to the handle so she can pull the straps up for her toys if she dropped them. It has enough room to carry enough supplies for her. It has a sun shade which was great. She frequently fell asleep while we were walking around. We could take the pack off with her in it and sit it on the floor, it stands by itself and can act as a high chair or we just left her in it to sleep. It is easy to adjust. At the end of the day, I would rather carry her than a hiking specific pack carrying other items for the day that was 15 pounds less. We purchased the rain ponch and it kept her dry when it rained. We used it on both hiking trails up in the Alps, walking in cities and Venice. We prepared her by carrying her around months before our trip so she was used to it.

We loved it and have HIGHLY recommended it to anyone in who will listen. My wife also uses it to carry her around. She is now 3 and still uses it. If she gets tired, we put her in the pack rather than fighting with a stroller.

Thanks everyone!

mchpp, thanks for letting me know about the Jennerbahn - the English version didn't seem to say that when I checked.
csu15269 and TravelingMom, the Osprey is very expensive where we are but we do have a basic carrier so will probably use that. I've used it before, just not for hours at a time, so I don't know whether we'll be able to enjoy it!

After doing a bit more research, I've read that the boat ride to the Konigsee is quite "boring, slow (2 hours), and over crowded with tourists" and that St Gilgen/St Wolfgang might be much better. I'm hesitant because I've done so much research on Konigsee and know nothing about getting to St Gilgen/St Wolfgang. Any advice on whether the two are comparable in beauty, tourist crowds and ease of visiting?

Or would you recommend that we stretched out our time in Salzburg to two full days (though I heard that one full day is sufficient if we were not museum people)?

Thanks in advance!

Königssee is in a German National Park. Yes a lot of tourists but it is a National Park, nature. The electric boat ride round trip is about 2 hours to St. Bartholomä, but there is a show with horn playing and echo on the way. The route through Schönau am Königssee to the boats is very touristic and Kitsch. I like the natural beauty of this lake and as I'm not that far away I'll return. https://www.foto-webcam.eu/webcam/funtenseetauern/
St Gilgen/St Wolfgang Austria on the Wolfgangsee is very touristic especially St Wolfgang. You can tour the lake with the regularly scheduled tourist boats/ferry. I have never been on the Schafbergbahn, but something to consider. St Gilgen can be pricey. I never saw a Tesla car until last Fall in St Gilgen and I saw two parked within a block. No problem with a stroller other than some steps in these towns. http://www.bergfex.at/zwoelferhorn/webcams/c709/
If you are considering St Gilgen/St Wolfgang then you might also consider Hallstatt, AT and the salt mine there transportation dependent. Hallstatt is super touristic during daylight hours, but it's another well known destination outside of Salzburg and popular in these forum.

As I remember, the path from the end-of-the-lake boat dock at Salat to the Obersee is not very good for a rolling anything. There are big rocks and tree roots across the path.