H1 with one company, work for another?

[30K]

I hope somebody would help explain this to me. I would really appreciate it.
I am on H1 with a consultant company (Company A), who send us to other companies on projects.
Now I have an offer from a good company (Company B), willing to hire me, but they don't want any third party (consultants) between me and it. They also have reservations about H1 approval, as it is just a temporary project and not a permanent job.

Now I know that I am not allowed to work on the payroll of Company B, if I have H1 approved only for Company A.
THE CONSULTANTS (Company A) ARE WILLING TO ISSUE ME A LETTER STATING THAT THEY ARE ALLOWING ME TO GET PAID ON COMPANY B'S PAYROLL.

They are saying that this letter will allow me to work for this company B without me having H1 approved for B.

Do you guys think it can work?

[30K]

At one immigration website, I found:
"The new rules also affect contractors who move H1-B workers from job to job. A new "time test" allows employers to shift an H1-B worker to a worksite not listed on the LCA for 90 days without filing a new LCA."

Another website says:
"In general, the employee can take short-term assignments in a location which is not covered by a certified Labor Condition Application (LCA), as long as that assignment is for less than 90 days."

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It looks to me that it is possible to have H1b from one employer and work for another, if I work for 3 months only.

I would really appreciate if anyone could please confirm this.