They have had their firstborn christened, should they just ignore they have a second child and not christen her because there's an element of people who for some bizarre reason don't like it?
This keeps coming up, and I don't understand it. They had their firstborn christened in the church nearest their residence at the time. Doing so was a de facto requirement of their jobs (working royals) at that time. They now live thousands of miles away, and no longer have those jobs. Normal families who've moved such a long way between the births of their two children don't fly back to their previous residence just so DC2 can be christened in the same place as DC1. If they want Lilibet to be christened in the Anglican tradition, there are dozens of Episcopal churches within driving distance of their current residence. They may have already done so for all anyone knows.
*As the media are making up stories repeatedly, I assume you mean that Harry has to take legal action every time they do?
And if he doesn't, it must be true?*
When someone repeatedly threatens legal action over trivial things like who said what to whom about a baby name, then them letting a much more serious accusation pass without issue will reasonably be viewed as confirmation of its truth. That goes double when the threatened suit was over something inherently subjective (whether the Queen was "asked" or just "told") and the current issue is more factually provable or disprovable (whether the contract Harry signed was for one book or four books). That's why the "never complain, never explain" approach they hate has lasted this long. It's not because no one besides Harry or Meghan has ever wanted to deny or correct anything false in the newspapers. It's because they know that once they start down that road, there's no easy way to turn back.
Still waiting to hear which charity got the $20+ million advance.