(I've switched to a different kind of text Editor to compose in.
it doesn't capitalize the first letter of a sentence like the other one did, so I'll try to put in more new lines to make it more clear.)
Romans 10:2 For I bear them record that they have a zeal of God, but not according to knowledge.
"they have a zeal of God"
becomes
"they think have a zeal of God"
imo, and I mean this gently, that's a very loose interpretation!
to be fair then, I think the lumen gentium:
In the first place amongst these there are the Muslims, who, professing to hold the faith of Abraham, along with us adore the one and merciful God, who on the last day will judge mankind.
would be interpreted this way:
In the first place amongst these there are the Muslims, who, thinking they profess to hold the faith of Abraham, think they along with us adore the one and merciful God, who on the last day will judge mankind.
I know you try to defend catholic. But I don't agree with your interpretation.
You have to read Romans 10 and put it on the context
10 Brethren, my heart's desire and prayer to God for Israel is, that they might be saved.
2 For I bear them record that they have a zeal of God, but not according to knowledge.
3 For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God.
There are additional explanation that I consider to come to that interpretation
Zeal to God, but
1 first additional
But not according to knowledge, mean wrong
2. Second explanation
Ignorance of God righteousness, or lack of god righteousness mean wrong god
3. Thirth additional explanation
Not submitting unto the righteousness of god, mean not submitting to God
Zeal to true God but not submitting to God mean zeal to wrong God,
4 going to establish their own righteousness, to me like make their own god.
So I don't believe zeal to the true the true God doing such 4 things
My conclusion is not true God
Verse 4
4 For Christ is the end of the law for righteousness to every one that believeth.
To me mean the righteousness is Christ.
Not submit to Christ mean not submit to God or righteousness, or not adore to God.
Compare with ccc 841
Muslim and Christian adore the same god. That is pure lie