Learn Hardcore Japanese: tentyou ha isogasisou da kedo, ato de syoutaimeeru wo okuru to itta. - The manager looks busy, but he said he’ll send an invitation email later.

Breakdown of tenchou wa isogashisou da kedo, ato de shoutaimeeru o okuru to itta.

What does the particle は after 店長 do here? Why not が?

は marks the topic: “As for the store manager...”. It sets 店長 as the thing we’re talking about for the whole sentence.
Using は naturally lets the reader/listener assume the store manager is also the one who said the second clause (unless otherwise marked).
が marks a specific/new subject and often emphasizes it. 店長が忙しそうだ = “It’s the manager (as opposed to others) who seems busy.” With a following clause linked by けど, は is more natural to carry the topic into the second clause.
You could say 店長が忙しそうだけど、後で… and it’s not wrong, but it puts a bit more focus on who seems busy, and the subject continuity to the second clause feels a touch weaker unless you restate it.

Why use 忙しそう instead of 忙しい? What does 〜そう mean here?

忙しそう is the “appearance” そう: it means “looks/seems busy (from what I can see).”
忙しい states it as a fact (“is busy”). 忙しそう is softer and based on outward signs—more appropriate if you’re not certain or being tactful.
Note there is also a different そうだ meaning “I hear that...” (hearsay), but in this sentence it’s the appearance one.

Do I need だ before けど? Is 忙しそうけど okay? What about だけど vs だけど?

After this 〜そう (appearance), use the copula before けど: 忙しそうだ＋けど.
In writing/speech, this is usually fused as 忙しそうだけど (the だ is inside だけど). Saying 忙しそうけど (without だ) is generally not standard—avoid it.
Polite versions: 忙しそうですが, 忙しそうですけど.
Contrast: with a plain い-adjective you don’t need だ: 忙しいけど (because 忙しい is already an い-adjective).