Okay, I figured this out.
First, let me point out that what I was talking about is not for open and click tracking in general, but for a custom domain for that tracking.
Second, I found out that a CNAME record is only not allowed to have another record with the exact same name. The name of the TXT record for verification has the prefix "_amazonses.", so the name is not exactly the same. The record that was preventing me from adding the CNAME record was an SPF record (a TXT record with a value starting with "v=spf1"). That record had been added automatically by my server company in case I was going to use this subdomain for sending e-mails. But this subdomain is only for open and click tracking and will never be used for e-mail, so that SPF record is completely unnecessary. So I deleted the SPF record, and after doing that, I was able to add the CNAME record.
So I now have a valid custom domain for open and click tracking.
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