You don't give up any rights. Your company (which is a creature of statute) gets whatever rights the law gives it. Which given its privileged position (perpetual succession, limited liability etc.) are inevitably going to be different from those of actual people.
Which is fine. If you want to exercise your right to free speech you can. You can't necessarily make a company exercise those rights on your behalf because the company (in a fairly profound legal sense) isn't you and has limits by law.
We're not talking about how things legally are in US (especially considering that this isn't even a story about US!), but about what the proper handling of free speech vis a vis corporations should be.
And it's hard to not come to a conclusion that corporations, being legal creations of the state distinct from a mere assembly of individuals, cannot have any natural rights, since their very existence is a privilege; the society can attach whatever restrictions it wants to corporate charters, so long as they're uniform.
(Note that there's a separate question of what restrictions are actually a good idea. I would actually argue for the free speech side here, but that's beyond the point; the point is that, for corporations, either way, we're talking about granted privileges, not inherent rights.)
Separately, "Congress shall make no law", interpreted most literally, means that e.g. state legislatures can make such laws. The First Amendment was not incorporated against the states until 1925, so this country has only had federal free speech protections for less than a century.
Which is fine. If you want to exercise your right to free speech you can. You can't necessarily make a company exercise those rights on your behalf because the company (in a fairly profound legal sense) isn't you and has limits by law.