You're unlikely to get an upgrade via a cheapie offer if you just call in cold. The call centre will give you a price for a fare rules upgrade, which would cost you the change fee plus the applicable fare difference. Otherwise, BA wouldn't sell any premium tickets at the full price because people would always buy a cheapie ticket followed by a cheapie upgrade. So the price that you've been quoted probably is right. You have to remember that you're never entitled to a low-price upgrade, so you do need to set your expectations accordingly.
On your itinerary, you probably won't be offered a special price upgrade in advance, usually known as a POUG, which are discussed in this thread:
https://www.flyertalk.com/forum/brit...17-thread.html
If you check-in online, you may be offered a special price upgrade at that point during the check-in process. It's important to remember that if you are, you have to take it at the point at which you are offered it. If you don't take it then and complete your check-in, you're unlikely to get that offer again via the website.
If you don't get an offer during the online check-in process, the next port of call is to ask at the airport whether there is special offer upgrade that you can buy. These are usually known as an AUP, and they are discussed in this thread:
Airport Upgrades (AUP) promotion 2018 thread
Via all of these upgrade routes, if you are flying a BA-operated flight and collecting into BAEC, then you will be entitled to the Avios and Tier Points of the upgraded cabin, which should then put you over the threshold that you need. However, it is always possible that you will be automatically awarded at the rate for your original ticket, and that you have to ask for the correct amount afterwards.