I cannot provide API access, although I am loading the matrix rate shipping rates in a completely custom way - I have rewritten checkout entirely. If you are interested in loading shipping rates for a quote generated by the matrix rate module then I could answer that?
Edit with method for retrieving matrix rates:
1. Checkout Session Quote must have a shipping address that has been saved so it has an id.
2. This may be partially redundant but I set collect shipping rates on the quote.
3. I load the shipping rate from the shippingRatesCollection (on the shipping address from the quote, and I take the first item since we are only enabling matrix rate), I retrieve the code for that shipping rate (which has been generated based on the quote contents and price) I then set that shipping method code on the shipping address as this is how the quote calculates overall totals and shipping prices (based on shipping by default).
Here is a segment of the code:
// Hoping saving the cart (and therefore both address values will allow estimating methods)
// error handling for no delivery method available
$method = $delivery->getShippingRatesCollection()->getFirstItem();