Isn’t there a free speech aspect to canaries? Can the court force someone to publicly tell an explicit lie?

A NSL or similar already prevents you from saying something and limits your free speech.

The court will, either way, not be impressed by someone communicating that they got a warrant by not communicating in a previously arranged manner (this is basically communicating in a code language).

only if you're covert working for 3 lettered agencies can you lie and get away with it. Though they'd get you out some other way to avoid blowing your cover.

I would argue that you brought yourself in this mess. The only valid reason for having a canary is to evade the requirements of the law, so you can't complain.

It's an argument, I'm no fan of secret orders or the government telling you to lie or keep quiet.

Fundamental rights generally override "you got yourself into this mess", don't they?

Fundamental rights can be in conflict with each other. It's not uncommon for a judge to impose gag orders, which ostensibly protects a fair trial, but infringes on a person's rights to talk about a case in public.

Well, the right to freedom of movement can be taken away if you've taken yourself into a mess that ends you in prison (and in tens of other cases for that matter).