A few thoughts that might help you shape your plans:
To do this entire itinerary by rental car seems to me to be less than ideal. There are hardly any targets on your list where a train isn't a better solution. Trains are the way of getting around, in the cities where cars are a hindrance, and in the mountains where cars often aren't even allowed. Look at www.oebb.at for Austria, www.bahn.de for Germany (also for connecting trains to Austria and Switzerland), and www.rail.ch for Switzerland.
Rent a car wherever it is a better solution, for a day or three at a time.
In Austria your rental car has to display the Austrian road-tax sticker called Vignette, without which you are not allowed to drive on the main roads.
Once you enter Switzerland, your car has to display the Swiss version of the "Vignette" - you can buy it near the border (but you probably won't get reimbursed).
Germany has no such requirement.
If Vienna is too expensive to fly into, look for an alternative. Possibly Munich. If so, visit what you wish to see in the region as you first get there, possibly by rental car, but I suggest looking into the many other ways of getting to, say, Ludwig's Castle (lots of tour buses take you from Munich daily, hassle-free, with knowledgeable guides that maximize your experience.)
The "Romantic Road" is not really a road to speak of, it is the towns along that "road" that make it worthwhile, so read up on them, decide which you want to see and rent a car for a day or two, just for that.
You can visit Melk as a half-day excursion by train from Vienna, trains take an hour or less and it's a ten-minute walk from the station to the Stift.
You don't need a car in places like Lucerne and Lauterbrunnen, in fact, a car is a bother there.
Europe is full of budget airlines - if your luggage isn't excessive (check each airline's website carefully) you can save a bunch by picking the cheapest transatlantic flight to wherever there is one of those airlines to take you to your final destination. Which - as mentioned by previous poster - opens up all kinds of options. Avoid the UK, though - their departure tax will work against this strategy.
If Munich (MUC) also isn't the ideal arrival airport, maybe Zürich (ZRH)? Then turn your trip around. Or start out in Frankfurt am Main (FRA)? (Realize that there is another Frankfurt, not as important but nevertheless, it is called Frankfurt an der Oder).
Or Brussels? Or ??? Check those other airlines and their connections on www.whichbudget.com and on www.skyscanner.net, read the rules, and keep in mind that a late arrival from overseas does not oblige that connecting airline to honor your ticket if you miss their onward flight, so schedule accordingly.
If Vienna costs you 1200 bucks and Brussels 800, and a connecting flight 50 Euros, you're laughing. It happens.
Lauterbrunnen is shady, down in the valley, I wouldn't stay there. If you want to keep your options open about the weather etc., (not a bad strategy), stay in nearby Interlaken, it is a town on the flat between two lakes, with lots and lots of great targets within reach, in all directions, including Lauterbrunnen (20 minutes by car, parking severely restricted, a few minutes more by train).
Interlaken caters to visitors in a very efficient way - lots of hotels and restaurants and two train stations. You can hop on public boats on either lake, go up on (look these up!) the Niederhorn, the Niesen, the Brienzer Rothorn, and, if worse comes to worst and the weather totally shuts you down, hop on a train to Bern and enjoy the ancient inner-city center that is still a vital active city with the seat of the government and such - and the arcades that flank the pedestrian-only inner-city streets keep you dry while you gallivant.
After Switzerland I would suggest taking the train to Freiburg im Breisgau where you would have reserved a rental car to take you via the Black Forest to Frankfurt.
Hope this helps.