Do you intend to keep the same car for a full fortnight (i.e. renting in ARN and leaving the car in CPH)? Or renting a car in ARN and then again in BGO? A two-week one-way rental ARN-CPH in September is around USD 1200 for a C-class car; not too bad in my opinion, so it certainly is a viable option financially. But the ferry is OSL-CPH is an added cost (even though passenger fare is more expensive than the car fare). Gothenburg is a nice city, but I wouldn't go there instead of Oslo; Oslo has some offerings as a capital, which makes it more interesting than Gothenburg, unless you have a particular reason to visit the city. The journey ARN-OSL is underserved at the moment; too much LCC competition means having train connections on the direct line only at weekends. Daily train connections available via GOT, though. A possibility would be to do the journey ARN-GOT-OSL-BGO-OSL by train, using a ScanRail pass , either a 5/60 or a 8/60 or a 21/21 pass. The cost isn't that bad, and it can also be bought in Scandinavia (but then only valid for three days in country of purchase). It's a go as you like pass, but seats must be reserved on most trains. Typical cost is 288 Euros for a 8/60 pass per person. Most trains only have standard ("Second") class, which is very acceptable. Sleeping cars are available and are comparable in cost to a hotel. Having the pass also makes day trips possible at little extra cost.

A possible itinerary would be:
Day 1: US-ARN
Day 2: ARN
Day 3: ARN
Day 4: ARN-GOT
Day 5: GOT-OSL
Day 6: OSL
Day 7: OSL: Excursion to Northern Norway: TOS suggested
Day 8: OSL: Excursion to Northern Norway
Day 9: OSL-BGO
Day 10: BGO
Day 11: BGO (do Norway in a Nutshell one of these two days*)
Day 12: BGO-OSL (morning train): ferry for CPH leaves at 5PM from OSL
Day 13: Arr: CPH
Day 14: CPH
Day 15: CPH-US

If not using a ScanRail pass for day trips a 5/60 pass, would cover this (or you can drop using it on the Arlanda Express on the arrival day to save a day's use) I would take a night train with a sleeper in one direction OSL-BGO; e.g going directly from OSL airport to the RR station and take the night train.

If going by day both ways between Bergen and Oslo, the NIAN can be incorporated in the BGO-OSL or OSL-BGO run on certain days. For flights OSL-Northern Norway (Tromsų (TOS) is probably the best for shorter trips for first timers!) check both SK and DY (Norwegian a LCC) for tickets. Try and take boat excursions. It might be late for tourist trips, but trips on scheduled journey are available at OSL, ARN (the Stockholm archipelago is famous), BGO (the best of which is part of the NIAN), TOS and CPH. For train timings, check www.bahn.de. This official site of the German Railways is recommended also by the Norwegian State Railways, and has most connections in Europe listed.

The above is a viable schedule, and of course can be adjusted, also to include other places. To get more time in CPH, I would cut a day in OSL or ARN.