TL:DR 300MHz isn't really that fast. Signal integrity will be fine, but EMC may be an issue.
Your looking at 300MHz, which isn't that fast. Assuming you're looking at single ended signals, rather than LVDS, and you have some reasonable checks in place on the communication, I can't see why you need to worry about a 6 layer for that, a four layer would do the job (for SI, clearly you may want 6 or more layers for routing).
You talk about putting ground references top and bottom on the signal. There is no need to do that, only one layer will be the reference. With good design you should be able to work out which it is, and so reduce layer count to 4.
The wording of your question makes me think you are looking at this purely from a SI point of view. While SI is important, the main reason to consider the reference plane is EMC. Without considering your reference planes it's easy to design even a slow(ish) PCB which radiates massive amounts of EMI, and thus becomes unsalable in many markets (different areas have different radiated emissions limits). If you are doing this for a home project, EMC issues can be ignored, or at least left alone for now. If you are making a product to sell, then it IS a problem.
Don't forget that the signals will still be using a reference plane when they go through the connector, and that can completely muck up your plan.
Second "don't forget": if the distances on these signals on the PCB are short (relative to their frequencies) then any kind of reference planes is irrelevant.
You are worrying about this more than you need to from the sound of things. I would recommend an EMC design course to clarify these points (I've been on a couple in the UK, both were useful).