In your example, the word ‘so’ is, despite having been moved to the head of the clause, the subject predicate in the clause. It stands in for ‘fine’—in a way, the sentence is really, “fine will you [be]” (thus speaketh Yoda).
The underlying issue here is that there is a difference between how you can elide the copula (the verb ‘to be’) and all other verbs when you have two clauses where the verb from the first is repeated in the second.
When you elide the verb, what you actually do is replace it with either a dummy verb (‘to do’ in English), a modal verb, or an auxiliary verb. The main verb is removed, and only the dummy/modal/auxiliary is retained. (The version with a dummy verb is not possible with the copula, modal verbs, and auxiliary verbs themselves, they have to be repeated.)
With active verbs, even if the constructions in the first and second clauses are different (and use different forms of the main verb), you can still elide the main verb from the second clause and have only the dummy, modal, or auxiliary verb left in its place, carrying all information about what from the main verb would be in if it were present. So you can say both, “I kicked the ball, and so did he” (replacing with dummy verb), “I have kicked the ball, and so has he” (auxiliary only), “I can kick the ball, and so can he” (modal only), but you can also mix between these three types: “I kicked the ball, and so can you” (‘kicked’ is simple past tense, but if unelided, the second clause would have ‘can kick’ with the main verb in the infinitive—but you can still elide it).
This latter bit is not possible with the copula verb ‘to be’: you have to have parallel forms in the elided and non-elided parts. If you have a past tense ‘was’ in the first clause, for example, you cannot in the second clause have a future tense ‘will be’ and then still elide the main verb, leaving only ‘will’ behind. Only if the main verb (‘to be’) is in the same form in both clauses can it be elided: “I have been a student for five years, and so have you” (have been + have been); “I could be strict with him sometimes, and now so must you” (could be + must be—different modals, but both modals that take an infinitive of the verb); but not “I have been strict with him, *and now so must you” (have been + must be—mismatch, does not work).
Therefore, in your example, since you have a past tense copula in the first clause, but a future tense copula in the second clause, you have to retain the entire copula in there:
I was fine. So will you be.
