I have no clear answer to you yet I can give you some insights, my understanding of this.
First, I am not sure that nodetool will let you "rebuild" from the DC the node is in.
Then this would only work properly (if it works) because you have 3 nodes and a RF of 2 or 3 and so all the data is already present in your new DC, else you will rebuild from an incomplete DC. BTW Consistency Level - quorum - has no impact as CL is for clients and you are on server operations, what matters here is the RF and what data each node "hold". Using 'repair' or copy SSTable directly instead of rebuild are options you might want to consider (in the case all your data is already present in DC2 with only 2 nodes loaded).
That was to answer to your question, but I would say you should stick with the procedure, it should definitely work, you just did it twice... "as we'd like- streaming in the new DC would make things faster and ease some headaches." being creative and deviate from standard procedure sometimes work great... But often increase headaches and make things slower, take care, be sure of what you're doing or follow procedures, imho.
"Our DC's are linked by a VPN that doesn't have as big of a pipe" --> you rather should try to solve this as much as possible, you will need to repair your cluster which can be quite bandwidth consuming.
As global advices for new DC: you might also want to disable read_repair_chance on your tables to avoid cross DC at read time (use dclocal_read_repair_chance instead), use "Local_Quorum" instead of quorum and have your clients sticking the local (to them) DC.
Hope this will help, even if I can't answer precisely the "would it work" question.