How to buy Slovakia discounted international rail tickets outside Slovakia?

I live and work in Bratislava, and have bought (or tried to buy) international train tickets on numerous occasions while I've been here, including for this route. You're right to say, as a starting point, that tickets from Slovakia to other countries generally cannot be bought online via the Slovak Rail website. Instead, Slovak Rail say you should buy these tickets in person from a station with a Complex Ticket booth (which includes Bratislava main station).

In answer to your questions:

Is there another way to buy the offer in advance (pay in advance, pick up on day of travel)? Maybe another ticket office / tourist bureau in Slovakia? (I couldn't find any)

The Man in Seat 61 suggests not, and that the only alternative is to purchase the tickets over the phone (which I see you've already tried). I'm not aware of any agencies here in Slovakia, although I suspect some must exist. I also tried The Trainline's European site, but again found that the tickets were not available for purchase - and I'm not convinced they're the cheapest for tickets from Bratislava at any rate.

Are you aware of any other countries where I can buy discounted tickets from Slovakia? (This was proposed by the Slovak Rail phone line, but I called in Switzerland and Hungary: not possible)

Yes - it looks like you can use this site, which is a Polish rail ticket agency. The site is in English and sells tickets for trains from Bratislava to Warsaw. They offer international ticket delivery. When I put in delivery details for the UK, as a test, it stated that the ticket would be a print-at-home e-ticket, so there may be no need to pay for delivery costs.

I had two caveats on this, the first being that I have no idea how reputable Polrail are, and the second being that I couldn't find the discounted €19 price, even looking around 2 months in advance. Thankfully, The Man in Seat 61 (see this page) has answers to both questions:

Alternatively, you can book through reliable booking agency www.polrail.com - their booking system is at http://booking.polrail.com. [...] Polrail always shows full price, but tell me they will offer the cheap advance-purchase fares if they find them available.

If that doesn't suit you for any reason, my suggested alternative is to book the Nightjet sleeper from Vienna to Warsaw through Austrian Railways (OEBB)'s site, which is here, and then get the train from Bratislava to Vienna. Advantages of this: the OEBB site is easy to use and allows you to print tickets at home (and it's in English), you can buy the whole route online via the site, it's not a third party site, and it should cost around €40 all told (for a seat; a couchette would cost around €60). Day trains from Vienna to Warsaw are also available, but looking around a month in advance, they cost in the region of €80.

I accept that this option isn't the €19 train, but it's still cheaper than the €80 option.

Last option: I know the same offer (19 EUR), from Hungary works, as well. I could buy and have a ticket from Hungary, but only board the train in Bratislava: would my ticket be valid?

I don't honestly know the answer to this, but seeing as you have a valid train ticket that covers the entire route, I don't see why not.