You can ignore the stuff from the CEO, yes. But once you get a letter from a lawyer you should at least have a lawyer look it over and possibly draft a response.

It can sometimes hard to tell frivolous vs. serious letters, and ignoring a serious one is not a good idea. Given any sort of emotional attachment to your business, and lack of legal experience, you might not be the best person to judge the situation.

What? Failure to heed a legal threat can result in a lawsuit. Failure to appear can result in a default judgement against you. Appearing in court without adequate representation isn't much safer. Appearing with adequate representation is probably out of OP's budget.

Ignoring the posturing of lawyers on the other hand can be plenty expensive.

That part yes, but what do you do once you get served? What do you do when you need to respond in court?

The point is that aggressive litigation is a common tactic large companies use to bully smaller players, because they know that the smaller players don't have the resources for extended legal battles.