Couple of Notes/commentary Franze (that will also help anybody else with this stuff)
Your loadout is really not too heavy, The Hornet maneuvers quite well with 6 Sparrows and 2 Sidewinders, its only an extra 3,000 pounds, and the A/A missiles do not add a whole lot of drag (should be about the same as a clean jet with 1 full tank in regards to weight and probly better in drag).
What was making you have to use the G limiter override was your Speed
Corner speed for the hornet is going to be ~350-400 knots, you are merging with the fulcrum in your 2nd screenie up there at 550+ knots.
This is going to make your turn radius the size of Texas and you are on the power side of the curve at that speed and cannot pull enough AOA to start bleeding down your speed without immediately reducing the throttle.
Next, The reason your sparrow missed (initially) is because you fired it at 20NM from 7,000 feet
The sparrow is a perfectly capable 20NM weapon in DCS if fired from 20-30,000 feet or greater (if you get to 40,000 you can shoot it from 25 without too much trouble).
If the bandits are down low as well, your WEZ is only going to get smaller due to the thicker air the missile has to travel through.
A generally decent rule of thumb for Sparrows for head on shots (and amraams btw) is that from 10,000-20,000 feet, your Max WEZ is about the same as your altitude, plus a bit if hes significantly higher than you, minus a bit if hes significantly lower than you. Below 10,000 10NM works pretty solidly all the way down. Above 20,000 for sparrows you have to get all the way up to 40,000 to start seeing a significant enough range improvement. This is due to the fact that sparrows only function in game (assumed battery timer) for 60 seconds, you have to get a lot higher for that 60 seconds to start translating into extra range, you will get better kinematics out of a 30,000 vs 20,000 foot shot however.
Amraams have a longer time to live ~(90 seconds IIRC) so you can take shots farther out with them.
If you had been up at 20,000+ feet when you took your first shot, it probly would have connected.
Also a good rule of thumb on these types of intercepts. If you don’t crank (put him 45-50+degrees off your nose) after the shot, expect the missile to timeout (hit or miss) at about 1/3rd the distance you shot it (in your case a 20NM shot, you should expect it to hit/miss at 7NM from you) If you crank, its 1/2 the distance from you (10NM in this case)
IF THE BANDIT REMAINS POINTED AT YOU (if he turns around these rules of thumb are invalid).
You had plenty of time for a follow up shot if you see 7NM and no explosion, go ahead and point at him take a second sparrow, and know that it is going to timeout, right about when you are getting ready to start employing a sidewinder, giving you another swing at killing him prior to the merge.