There are two questions here: whether the sentence is unusual or ungrammatical on the one hand, and what sort of a clause it is. Here’s the sentence of your concern as found in your citation:
- I have that which I should have told you summers ago.
The reason is sounds a bit funny is because things you tell someone are normally abstractions like opinions or situations, positions or tales, secrets or truths. Therefore they are not in the same category as tangible objects that you can keep in your possession, to “have and to hold” as the saying goes.
If this were a tangible item, adding an about would help a little:
- I have that which I should have told about you summers ago.
The text following that sentence suggests that what he’s talking about here is a secret. That’s what’s unusual here. A more expected formulation might be:
- There is something (that) I should have told you (about) summers ago.
- I’ve got something (that) I should have told you about summers ago.
Intangibles like truths and tales are more often known than possessed. For example
- I know that which I should have told you summers ago.
In any case, the clause beginning with that which serves as the direct object of the main verb. Under most modern analysis, we would not call this instance of that a relative pronoun, let alone a demonstrative one. Demonstratives are words like this, these, that, those. They can be used in front of a noun as a demonstrative determiner taking the place of the definite article’s slot in the noun phrase, or they can be used by themselves substantively as standalone demonstrative pronouns.
Perhaps there’s an argument to be made that is a pronominal that standing in for the thing. That argument would be strengthen by showing that a different demonstrative pronoun could have been used here, like those standing in for a plural the things or the people.
- I know (of) those things which I should have told (about) you summers ago.
- I know those which/who(m) you’re talking about.
But another way to look at that which is for that to be a complementizer like whether and if can be. It alternates with the thing (that/which) if definite, and a thing or something (that/which) if indefinite. More commonly the heavier formulation the thing that/which gets replaced by a simple what to create an embedded wh‑ clause:
- I know that which you need.
- I know the thing that (or which) you need.
- I know what you need.
When it’s not things that but rather people that/who(m), we can use who(m) instead of what:
- I know those which you need.
- I know those whom you need.
- I know the people that (or who[m]) you need.
- I know someone (who[m], that, which) you need.
- I know who(m) you need.
Embedded wh‑ clauses like these are considered embedded interrogative clauses under a small set of possible predicates including know, but are held to be fused relatives in most other cases. If it’s a fused relative, you can add ‑ever to what to say whatever for things:
- I can recommend that which you need.
- I can recommend the thing that you need.
- I can recommend what you need.
- I can recommend whatever you need.
Or for people when using who(m), add ‑ever to who(m) to make it who(m)ever*:
- I can recommend the people that/who(m) you need.
- I can recommend who(m)ever you need.
When there are several possible equivalent ways of saying the very same thing as we have here, longer versions often feel heavier and more cumbersome than shorter versions. They’re probably less common and so may sound more formal. This makes the choice of something (that) or that which sound less casual and more unusual, especially that which.
This may nonetheless have been a deliberate choice by the author, who may have had reasons for placing a heavier or less usual expression in the mouth of this character in particular. That’s something we can’t really address without getting into literary analysis, which would not only require knowing much more about the piece that your citation comes from, it would also be off-topic for our site.