Harry and Meghan knew the risk when they left the royals
The move to leave the Royal family was momentous, and not a decision taken lightly. The couple knew the possible outcomes when they chose to renounce their status. They should have anticipated paying their own security costs.
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The Argument
It's one thing if Harry and Meghan were stripped of their royal titles for no reason, but another when they willingly give them up. No one forced the couple to leave the royal family. In fact, it appears that the decision upset the Queen.[1]
The Duke and Duchess should have known what to expect when they dropped all of their duties and titles. Having grown up under the veil of royalty, Harry should know that the rest of the world does not enjoy the protection, celebrity, and privilege that his birthright grants. The same goes for Meghan, as she married into a very well-known and public environment, where it should come as no surprise that they would be subject to media and public scrutiny.
It's no secret that the royals live in a very unique context. It should come as no shock that they would lose some of the privileges of royalty, after deciding to leave their duties and titles behind.
Counter arguments
The tabloid's obsession with Harry and Meghan made it impossible for them to remain a part of the royal family. The public scrutiny—and at times, prejudice—directed at the couple meant that they needed to leave their royal status behind. They may know that means a loss of privileges, but it can hardly be suggested that requiring security is a privilege.
No, average celebrities do not receive public funding for security, but the average celebrity is not also a member of a national government. Through their royal ties, Harry and Meghan's status is different than most celebrities, and therefore should be treated differently.
Proponents
Premises
[P1] It was a voluntary decision to leave their royal status.
[P2] The couple knows that private citizens do not enjoy the same privileges as royalty.
Rejecting the premises
[Rejecting P1] The media and public scrutiny of the couple forced their hand.
[Rejecting P2] They had no choice in their status, as it was Harry's birthright.