How to use Format in TeXForm keeping expression as shown in Format?

Question

I have expression expr = int[1 - x + x^2 - x^3 + x^4 - x^5, x] and I want to convert it to Latex. But want to replace int by Integrate but without evaluating it.

I tried using Format and that works inside Mathematica. It prints correctly on the screen.

But when using TeXForm on expr it ignored the Format.

Here is MWE

ClearAll[int, x]
Format[int[e_,x_],StandardForm] := Defer[Integrate[e, x]]
expr = int[1-x+x^2-x^3+x^4-x^5,x]

And that what I want in Latex. But when now I do

TeXForm[expr]

it gives

"\text{int}\left(-x^5+x^4-x^3+x^2-x+1,x\right)"

Which compiles as

Instead, I wanted the latex be generated based on the display it shows on the screen from Format which should be something like

"\int { \left(-x^5+x^4-x^3+x^2-x+1 \right) \, dx }"

so it matches what it shows above on the screen.

I looked at this post Format and TeXForm does not work as expected and tried few answers there, but so far could not make anything work. It is almost 10 years old and many things changed and I was lost of the complexity of answers and what I am supposed to do.

All the expressions I wan to apply this to have the form

 int[ .... , x] 

Does there exist a simple solution to this problem?

I thought about just parsing int[..., x] myself to pick up the ... and the x using pattern matching, and then converting everything manually to Defer[ Integrate[...,x]] and then I can do

 TeXForm[Defer[Integrate[ 1-x+x^2-x^3+x^4-x^5 , x]]]

And this now works. it gives

 \int \left(-x^5+x^4-x^3+x^2-x+1\right) \, dx

But was hoping to avoid parsing it myself and let Format do it if possible.

V 13.3.1 on windows 10