English grammar, not sure?

Namely, I'm writing an essay for English and I wanted to write that Jem begins to boast fearlessly that he even dared to touch the house. Context does not matter in the case. He wanted me to do the whole thing:

Jem starts to boast fearlessly about him being good enough to touch the house

translate, but since that happened in the past, I thought it must mean:

Jem starts to boast fearlessly about him having done well to touch the house.

However, this does not recognize my laptop & I want the essay to be perfect. Can someone tell me if the version is correct or if not, as I can express the whole thing in the past, without forming a relative clause, so who / that, because I find the above version more beautiful.

Both is possible.

In the present it sounds more general.

Okay thanks.

Your sentence structure is capable of improvement. That bothers him and is superfluous. It would be better:

Jim starts to boast fearlessly about being brave enough to even touch the house.

With "have touched", it's clear that it's about the past.

Jem starts to boast fearlessly about him having even been good enough to touch the house.

Even though modern English often uses him (= gerund with his own subject) - see also: • Would you mind me opening the window? -

Personally, I prefer to be here (with a possessive lens) - see also: • Would you mind my opening the window?

You can also omit him or His here completely.