Can I replace the modal verb ‘must’ with ‘have to’ in this sentence? Does the meaning stay the same[Image]

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Yes you can. Simple answer. 'Must' has a very slightly different connotation to 'have to', but you certainly can.

Answer

Yes, you could use “have to” and it would still make sense. 
However, it would sound more modern. “Pollyanna” was written in 1913.

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We need to see more of the text to be able to help you.

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In modern North American usage, "have to" and "must" are equivalent. I can't addresss British usage.

English Grammar in Use by Murphy (5th edition, unit 31) says that "have to" is for factual obligations and "have to" and "must" can be used for perceived obligations.

excerpt
unit 31A 
I have to do something = it is necessary to do it, I am obliged to do it.

unit 31B
Must is similar to have to. You can say:
It's later than I thought. I must go (or) I have to go.

You can use must or have to when you give your opinion (for example, to say what you think is necessary...).

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