So I ended up doing my second alternative: book online in the desired booking class on another flight, then call in to change to the desired flight. I didn't have to pay anything extra.

For the benefit of others, here's a write-up of what I did when the SN booking engine wouldn't offer me the booking class I wanted, although it had inventory available:

Booking a specific booking class on SN
* First, go to ITA matrix and price the fare for the desired booking class on the flight you want. Note the price and the fare basis that ITA selected for you.
* Use ExpertFlyer or similar to find a flight with the same itinerary that has inventory in your desired booking class but no inventory in the booking classes immediately below your desired booking class. Alternatively, you could search the SN booking engine for a flight, but this will require you to check every date manually. To make things easier, make sure the flight is pretty close to the flight you want, as taxes and fare availability may change from one week to another.
* Search SN's booking engine for the flight you found on ExpertFlyer. The engine should offer you the desired booking class as there is no availability in the cheaper booking classes. Make sure that the price is the same as ITA quotes, less the 16 EUR "YR" surcharge that isn't applicable when you book through SN. Make sure the engine uses the same fare basis as ITA (click on "fare rules", it will show the fare basis).
* When both the price (-16 EUR) and the fare basis match with ITA, you can go ahead and book the flight on SN's booking engine. Seconds later an email will come in with a booking reference and ticket number(s).
* Call SN, give them the reference and ask them to change your flight to the flight you want. Since the itinerary, the fare basis and the price are all going to be the same, this is going to be pretty easy for the agent, with no add/collect or refunds to process. Ask the agent to keep the booking class. Note that the agent I called with wasn't going to change it anyway, even with a cheaper booking class open on the flight I wanted.
* Ask the agent to send you the ticket by email again and thank him/her.
* Check that agent revalidated your ticket (!) which is confirmed when you get the email with new ticket details - make sure everything is correct. If you don't get an email within an hour, call in again and ask to revalidate the ticket. Don't wait until the next day! Don't trust verbal confirmations that your ticket has been revalidated, this is an immediate cause for HUACA! You *need* to get the revalidated ticket, or you'll be at the airport and find out your ticket is "canceled" because you "didn't show up" for the flight you originally booked before you called in to change.
* After you get your revalidated ticket, go to the SN website, enter your booking reference and check that the reservation details are correct. Assign a seat on your flight. If you can assign a seat without issues, you should be all set. If it doesn't work, call in again and ask why you can't assign a seat online.

Alternative method ("upfaring")
The previous method will only be free of charge when the fare you want doesn't carry a charge for changes. If you don't want to book a so-called "flexible" fare, you can get a higher booking class by upfaring. This is very similar to the above, but in this case, you would make your initial reservation online on the desired date, and not in the desired booking class, and then call in and ask to upfare to the desired booking class. The ticket will also need to be exchanged and reissued instead of merely being revalidated. Upfaring *is* also a change, but I've read reports that SN waives the change fee in case of upfaring, as is the case with most other carriers. You will be required to pay the fare difference though. Upfaring isn't done so often, so agents may not be sure how to handle one, don't hesitate to HUACA if necessary.