A grantor trust is a separate legal entity; that is its primary function. However, it may be disregarded for income tax purposes only and is, thus, allowed to use the SSN of one of the grantors.
I do not understand whether you are encountering a systems problem that will not allow you to do this or a persuasion problem with a co-worker. If it's the former, you will need to sort this out with your core provider.
If it's a persuasion problem, then it's important that the people understand there are a number of circumstances where an entity can be "disregarded" for income tax purposes and use the TIN of a different entity. For example, a single member LLC where that member is a partnership or a corporation can use the partnership or corporation's EIN. Also, a corporation may have a number of separate corporate subsidiaries, all using the "parent's" EIN.
The same is true for a grantor trust. It can file a separate income tax return from the grantor or it may just provide the grantor with a schedule of its income. Then, the grantor is responsible for declaring and paying taxes on the income. In that circumstance, the trust may use the grantor's SSN.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.