I read OP’s post as the relative framing the interaction as a demand for an apology in response to her sharing that she was upset by their reaction.
Assuming it went as follows 1) OP asked the relative to look after her child, explaining their was no pressure as she had an alternative option 2) relative suggested they had no option to say yes 3) OP expressed her upset at this reaction, the upset arising from the relative viewing her request as unreasonable 4) the relative is now stating the OP was unreasonable for expressing her upset, framing this as a demand for an apology.
Zooming out, it sounds like both parties are expressing upset/ anger. OP is upset that she has been misunderstood and judged harshly despite having made an effort to approach the relative in a pressure free way. The relative seems to be feeling taken for granted, pressurised or coerced.
OP, from what you’ve posted, you have approached the relative in a thoughtful and respectful way. It seems likely that their response is influenced by other baggage which may or may not be connected to you.E.g. they may find it hard saying no and feel pressured simply because of being asked, they may have a long history or recent significant event of being expected to go along with requests and there bring fall out when not, they may ‘have beef’ with you about something else and this is leaking into this request.
I suggest you take up the offer from your friend, focus on your big appointment, and consider how to proceed with the relative once your event is over