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.

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).

Shouldn’t it be 忙しそうだったけど since 言った is past?

Use 忙しそうだったけど if your observation (“looked busy”) is anchored in the past at the time he spoke: “He looked busy, but he said...”
忙しそうだけど is fine if you’re observing now (he still seems busy as you speak) while reporting what he already said: “He seems busy, but he said...”
So choose だ/だった based on when the “seeming busy” applies.

What does the particle と do in 送ると言った?

Why is 送る in nonpast when 言った is past? Shouldn’t it be 送った?

Japanese tense inside quotes is relative to the time of saying. 送る here means “will send (later)” relative to when he said it.
If he had said “I sent it,” you’d quote 送ったと言った. But with 後で (“later”), the quoted action is clearly future from that past moment, so 送る is correct.

Who is the subject of 言った and 送る? There’s no “he.”

Japanese often omits obvious subjects. Because 店長は is the topic, the default interpretation is “the manager said (that he will send...).”
If you needed to switch subjects, you’d mark it: e.g., 店長は忙しそうだけど、鈴木さんが後で送ると言った.

What’s the nuance difference between けど and が here? Can I use が?

What does 後で mean, and how is it different from 後に / その後 / 後ほど?

後で (あとで): “later” (from now or from the relevant time). Very common, neutral.
後に: “after,” often used with a reference point: 会議の後に. Slightly more formal or written.
その後: “after that/thereafter.”
後ほど: polite “later” (businesslike): 後ほどご連絡します.

Why is を used with 招待メール? How do I mark the recipient?

を marks the direct object (what is being sent): 招待メールを送る.
The recipient takes に: 私に/田中さんに招待メールを送る. You can also say メールアドレスに送る. へ works for destinations but に is standard for recipients.

Is 招待メール natural? What about 招待のメール or 招待状?

招待メール is natural and common (compound noun). 招待のメール is also fine, just a bit more explanatory.
招待状 is a (usually physical) invitation letter/card. For paper or formal mailed invites, use 招待状. For email, 招待メール (or 案内メール for an informational/notice email) is typical.

Can I say 送るって言った instead of 送ると言った? What about dropping と?

って is a casual quotative that often replaces と in speech: 送るって言った is very natural in conversation.
Don’t drop the quotative entirely: 送る言った is ungrammatical. Use と (neutral) or って (casual).
You can also hear 送るって with the following 言った/言ってた implied from context in very casual speech, but learners should include the verb.

How would I say this more politely or with respect to the manager?

Polite: 店長は忙しそうですが、後で招待メールを送ると言いました。
Honorific (because the subject is the manager): 店長は忙しそうですが、後で招待メールを送るとおっしゃいました。
If he’s sending it to you and you want the benefactive nuance: …招待メールを送ってくださるとおっしゃいました。

How do I read the key words?

Could I add a benefactive feel like “send it to me”?

Yes. Say 送ってくれると言った (casual) or 送ってくださるとおっしゃいました (polite/honorific) to show it’s for your benefit.
Example: 店長は忙しそうだけど、後で招待メールを送ってくれると言った。

Can けど be used just to soften a statement?

Yes. けど can start a sentence to cushion what follows or end a sentence to soften it:
- 忙しそうですけど、少しお時間ありますか。 (softened request)
- 忙しそうなんですけど… (trail off, inviting a response)
In your sentence it links two clauses (“although/but”).

How do I say “doesn’t seem busy” with 〜そう?

For い-adjectives, use the negative of ある/ない pattern with そう:
- 忙しくなさそうだ = “doesn’t seem busy.”
- Casual: 忙しそうじゃない; formal: 忙しそうではない.
Don’t say 忙しそうにない (that pattern is for verbs: できそうにない = “doesn’t seem likely to be able to do”).

Can I move 後で elsewhere in the sentence?

Yes. Time adverbs are flexible:
- 後で招待メールを送ると言った (your version)
- 招待メールを後で送ると言った
Both are fine. Putting 後で earlier is very common; placing it right before 送る emphasizes the timing of the sending.