SharePoint Rest API - Set Lookup field when [Lookup Name]Id value is already in use - rest

Question

I have a SharePoint list with a number field that has an internal name of [ProjectId]. I later created a lookup column with an internal name of [Project]. I need to use a REST API to update the lookup column value. However, when I use
"ProjectId": 403
It sets the number field. I know SharePoint will append a 0 to a duplicate internal name. However, in this case it was created by the system. I can expand the Lookup column using "Project/Id" in a GET call. I just can't set it.
Does anyone know how to reference the lookup id in this case?
Here is the full code:
// basic patch items
$.ajax({
url: _spPageContextInfo.webAbsoluteUrl + "/_api/lists/GetByTitle('My List')/items(1)",
headers: {
"Accept": "application/json;odata=verbose",
"Content-Type": "application/json;odata=verbose",
"IF-MATCH": "*",
"X-RequestDigest": $("#__REQUESTDIGEST").val(),
},
data: JSON.stringify({
"__metadata": {
"type": "SP.Data.MyListListItem"
},
"ProjectId": 403
}),
method: 'PATCH',
success: function (data) {
console.log("success");
console.log(data);
},
error: function (data) {
console.log("error");
console.log(data);
}
});
I've tried these as well:
"Project0Id": 403
"ProjectId0": 403
It just returns an error saying the column doesn't exist.