It sounds like you’re actually conflating The Iranian Hostage Crisis and The Iran Contra Affair
Which is a very common mistake, you’re not the only one, both involve hostages and Iran, and they’re not totally unrelated.
This is all greatly simplified of course.
The Hostage Crisis was was basically part of the Iranian Revolution when Ayatollah Khomeini seized power, where a bunch of supporters of the revolution took hostages at the US embassy. This happened under Carter. Carter placed an arms embargo against Iran in response. And like you said they held off on releasing the hostages until Reagan was elected. The Algiers Accord that formally ended the crisis did not involve any arms sales.
Iran Contra happened under Reagan. Reagan wanted to fund anti-communist rebels in Nicaragua (the Contras) but was blocked from doing so by Congress.
Meanwhile, the Iran-Iraq war is going on, Iran is still under that arms embargo, and really needs some weapons for that war.
At the same time, over in Lebanon, Hezbollah (who Iran has some influence over) takes some US hostages.
So Reagan and his cronies, like Ollie North, come up with an idea.
To raise money off-the-books to send to the Contras, they secretly sell weapons to Iran, and in exchange Iran will use its connections to Hezbollah to get those hostages released (and the hostages were in fact released, but Hezbollah pretty much immediately took more hostages, this was in the middle of a whole decade-long hostage Crisis in Lebanon with hostages from several countries taken, this was really just a small part of that)
In my opinion, the lyrics tell a pretty straightforward story, no real need for metaphor
Someone releases a bunch of balloons into the sky, they get mistaken for enemy aircraft/missiles, and set off a war.
If you get more out of it with your interpretation, more power to you, but it feels like a bit of an unnecessary stretch though.