> Should I forward the letter by removing the personal/confidential
> details?
> 
> They also said they may want to know the details of the company when
> close to an arrangement.

By doing this in stages, they're trying to manipulate you.

Don't give in. Be willing to walk away. 

If you give them the details, they will burn you to the other company (see this real life [example][1]) and the other company will rightly put you on their "never-hire" list. 

And your current employer may even renege on their agreement with you since they will have destroyed your leverage. 

If you really want to prove to them that you have an other offer, you only need to say:

"This letter is confidential and I just don't feel comfortable sharing it. You don't believe me, and that's fine. I don't need you to believe me."


  [1]: https://workplace.stackexchange.com/questions/148389/what-can-i-do-if-one-employer-used-offer-letter-from-first-company-against-me