This is a slightly unusual usage of くれる.
In the vast majority of cases, when a verb is suffixed with くれる it refers to a favour done for the speaker. This is quite a strict rule, to the point that it would be grammatically incorrect for a speaker to use くれる to refer to their own actions. However, since the speaker (the writer of the article) in this case is not involved in the situation, this can't be the case here.
As such, this must be the extended usage of くれる to refer to a favour done for someone who the speaker/writer is putting themselves in the position of. For instance, if you're talking about something a stranger did for your brother, you would often use くれた even though the favour wasn't done for you personally, because being close to your brother, you naturally empathise with him in this situation (and may be relating the story from your brother's perspective).
So then the question becomes "who is the writer empathising with in this sentence?", and the answer is clearly Mr Inoue due to a couple of points. Firstly, this whole paragraph is about Mr Inoue and is presented as an anecdote from his perspective, so it makes much more sense to put yourself in the position of Mr Inoue than the customer who was brought up afterwards. And secondly, as you pointed out, an あげる form was used earlier in the same sentence to refer to something Mr Inoue did, which would make no sense if the speaker was identifying with the customer (since あげる always refers to a favour for someone other than the speaker).
In addition to the above points, the use of たら also points to the customer being the one who 喜んでくれた. One of the unique functions of the たら form compared to other linking constructions is that it's often used to lead into something that the subject discovers in response to or as a result of another action (as in sentences such ドアを開けたら、家が燃えていた "I opened the door to discover the house on fire"). Or to put it a different way, the construction "Aしたら、B" can imply a subtle hint of suspense after the action "A", as the subject waits to find out the outcome "B".
This same nuance applies here - Mr Inoue performed the favour of 録音してあげる for the customer, and then the たら implies that slight suspense as he waited to find out the outcome of his favour (what the customer would think of it). Therefore the following action 喜んでくれた naturally has to refer to the customer's response.