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 and delay the process.
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) and the other company will rightly put you on their "never-hire" list. Or they will try to delay the negotiation and the final paperwork until your alternative offer expires. And they may ultimately renege on their agreement with you since your leverage will have evaporated.
If you really want to play this right, you need to say something like:
"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."
In other words, you need to stop trying to control what they think of you. Because by trying to control what they think of you, they're the ones actually controlling you.
Either they give what you want or they don't. You can't really control what they're going to choose. But the one thing you can do is behave as if you were impervious to their attempt to make you prove yourself to them. Because with a competitive offer from another company, even if they never see it, even if they don't believe it, it doesn't really matter. Deep down, you know that you're able to walk away.