Your Pictures are too good, and not the best way to do it anyway.
There's more to taking a picture of a planet in another system than just getting a big enough telescope. Here's a Link to the best pictures of exoplanets we can currently take, with the best technology. You'll notice the only "direct image" is a blurred red composite shot of a planet 4 times the size of Jupiter. To do better would require hellaciously better telescopes. We're talking on the order of 14+ kilometre lenses, not the 360 meters of your best pic (24 combined 15 meter mirrors). So can you make such a telescope with 90s/2000 tech?
Yes, but really no. You have a cold war still on to some extent. (even if the USSR calms down about being The Leader of the World and gets a rapprochmont of sorts with the US/NATO, that's not enough to lead to peace. Even when the USSR collapsed the US and Russia weren't exactly friendly) The space race was in part a prestige US vs USSR competition to be sure, but it went as far as it did because all the tech had military implications on earth. Rocketry, space stations, moon bases, all that craziness had real-world military applications. Spinning up huge mirrors in space.... doesn't. 99
The theory on how you'd build such massive mirrors and lenses in space existed in the 90s, but there's no reason for anybody to try it. The space shuttle and moon landings had military applications which helped spur their use. A giant telescope... not as much. Because....
You don't need a phenomenally expensive and costly and hard-to-build visual telescope. Let's grant your 1985 results. I doubt you'd get the image you posted, but detecting a wobble and thinking "a-ha, there may be a roughly earth-sized thing in the goldilocks zone!" doesn't seem too far-fethced.
So now what? Well depending on how you want things to unfold you could have them do some exoplanet spectography which determines a lot of water and CO2 doesn't seem hugely beyond the times. There were telescopes doing the same thing for starts since at least 1898. I don't have the dates of when we started doing it for exoplanets (earliest I found was a 2013 journal article which seems about right) but if we thought there was "another earth" as close as your exoplanet it might have spurred the tech/idea into occurring sooner.
However I think the quickest thing to do post-1985 would be to turn a radio telescope that way. From your last pictures it looks like there is a post-industrial revolution civilization on one of the planets, judging by the lights. Though radio/tv broadcasts washes out over fairly short distances, astronomically speaking, it should persist over 3-4 light years. Sentient Alien Life Confirmed, first contact 3-6 years away!
TL/DR Photographing earth-sized exoplanets to the detail of your photos is beyond our current tech, let alone your timeline. However determining that there could be an earth-sized planet via stellar "wobble" could be within the grasp of 80-90s tech, and a radio telescope of that era could likely detect alien broadcasts. 99