Bayesian updating, as a method of learning in general, is induction. It's trying to derive knowledge from data.

Only in a sense so broad that Popper can rightly be accused of the very same thing. Bayesians use experience to decide between competing hypotheses. That is the sort of "derive" that Bayesians do. But if that is "deriving", then Popper "derives". David Deutsch, who you know, says the following:

But, in reality, scientific theories are not ‘derived’ from anything. We do not read them in nature, nor does nature write them into us. They are guesses – bold conjectures. Human minds create them by rearranging, combining, altering and adding to existing ideas with the intention of improving upon them. We do not begin with ‘white paper’ at birth, but with inborn expectations and intentions and an innate ability to improve upon them using thought and experience. Experience is indeed essential to science, but its role is different from that supposed by empiricism. It is not the source from which theories are derived. Its main use is to choose between theories that have already been guessed. That is what ‘learning from experience’ is.

I direct you specifically to this sentence:

Experience is indeed essential to science, but its role is different from that supposed by empiricism. It is not the source from which theories are derived. Its main use is to choose between theories that have already been guessed.

This is what Bayesians do. Experience is what Bayesians use to choose between theories which have already been guessed. They do this using Bayes' Theorem. But look back at the first sentence of the passage:

But, in reality, scientific theories are not ‘derived’ from anything. 

Clearly, then, Deutsch does not consider using the data to choose between theories to be "deriving". But Bayesians use the data to choose between theories. Therefore, as Deutsch himself defines it, Bayesians are not "deriving".

The point of this criticism is that to even begin the Bayesian updating process you need probability estimates which are created unscientifically by making them up

Yes, the Bayesians make them up, but notice that Bayesians therefore are not trying to derive them from data - which was your initial criticism above. Moreover, this is not importantly different from a Popperian scientist making up conjectures to test. The Popperian scientist comes up with some conjectures, and then, as Deutsch says, he uses experimental data to "choose between theories that have already been guessed". How exactly does he do that? Typical data does not decisively falsify a hypothesis. There is, just for starters, the possibility of experimental error. So how does one really employ data to choose between competing hypotheses? Bayesians have an answer: they choose on the basis of how well the data fits each hypothesis, which they interpret to mean how probable the data is given the hypothesis. Whether he admits it or not, the Popperian scientist can't help but do something fundamentally the same. He has no choice but to deal with probabilities, because probabilities are all he has.

The Popperian scientist, then, chooses between theories that he has guessed on the basis of the data. Since the data, being uncertain, does not decisively refute either theory but is merely more, or less, probable given the theory, then the Popperian scientist has no choice but to deal with probabilities.  If the Popperian scientist chooses the theory that the data fits best, then he is in effect acting as a Bayesian who has assigned to his competing theories the same prior.

Probability estimates are not falsifiable.

Probability estimates are essentially the bookkeeping which Bayesians use to keep track of which things they've falsified, and which things they've partially falsified. At the time Popper wrote that, scientists had not yet figured out the rules for using probability correctly; the stuff he was criticizing really was wrong, but it wasn't the same stuff people use today.

How would you do it without Bayes' theorem?

Here's one way.

(It's subject to limitations that do not constrain the Bayesian approach, and as near as I can tell, is mathematically equivalent to a non-informative Bayesian approach when it is applicable, but the author's justification for his procedure is wholly non-Bayesian.)