What is the best way for 800m runners to go about gaining speed endurance (progression)

The best way is to get a coach.

Two more approaches:

(3) If you can find some low-key meetings, start doing 3 or 4 races instead of 1.

(4) The more traditional speed-endurance sessions move away from doing all reps the same length and into 'cut downs'. For example, once you can do a bunch of 300s at 800 pace, you start to 'push out' the longest rep and use long recoveries.

You need to -

Warm up really well (decent run, drills)

Some fast strides so that 800 pace is not a shock

Some 'aerobic warmup' e.g. run a couple of 400s in 72sec with a minute in between, so that heart is revved up too.

Run 500m at 800m pace; 8-10 min rest (aim for a "complete recovery");

Run 400m with 5-6 min rest after (you'll need it!)

Then a couple more 300s or 200s (500-600 jog rec).

Good long cool down,

Decent, slow recovery run the next day

You'll know when to stop, it's when you can't hold 800 pace any longer!

Second or third time, you push it out to 600m. This will be tough and that's why the warmup must be serious and the teammates really help.

If you can do 600 - 500 - 400 - 300 at 800m pace, you'll be a lot more comfortable racing an 800.

It's about progression. Repeat every couple of weeks, and try to push out the distances for which you can sustain your target pace. Thus it is quite literally 'speed endurance' training.

I've seen at least one 800m world record holder doing this in training multiple times and it's a staple session for many good 800m runners during early season.

It doesn't seem like option 2 would be developing speed endurance.

Option 1 sounds better, because you're starting with speed, then extending it.