I suggest you combine the two methods. Drill a hole near the traces and thread a thin bare wire through the board, lay it along the exposed traces by a few mm and solder it on both sides.
If you don't have such a wire handy, just strip a piece of stranded wire and use one of the strands
Nobody will notice unless they look very carefully, and it will be reliable.
It will be possible to drill these if you have a very rigid drill press and carbide bits. Back the PCB up with a sacrificial piece of laminate, get bright light and safety glasses and get very close to get the holes near the center. Use the highest spindle RPM your drill press is capable of (30,000 RPM is not too much) and feed slowly, especially when the bits break through.
If your drill press is not rigid enough or the bits are the slightest bit dull, they will break off and can go anywhere, so safety glasses are not optional.
If you use steel bits (use the best cobalt steel if you try this) they may work okay, but they'll tend to skate around on the pad and not be in the center. Replace them every 50 holes or so as they dull in the glass laminate.
You'll still have to find a way to solder them on the top or add additional jumper wires from top to bottom. If that thing is a terminal block, for example, you'll have to add ugly jumper wires on the bottom to a number of the pads. It won't be pretty but it might help you debug.
I doubt it's economical to get a machine shop to do this unless the boards are very, very expensive or you're willing to pay a lot of money to compress schedule.
In the future, look at your files with a gerber viewer that includes NC drill files, make a checklist for what files have to be with a given number of layers PCB, and (IMHO) never do business with this supplier again.
Best Answer
Ask your assembly house what is possible.
However, even if they can do it, it will be at a massive premium to the point that I would bet that HDI stacked microvia with a load more layers would be cheaper.
The issue is that you would need each layer to be drilled independently, and then lined up, they will get lots of duds and it will be a hugely more expensive and time consuming process then say a 8 layer simple or even an 8 layer with one or two layers of stacked microvia (Which is a standard process).
The mix of blind and buried is what makes it a real pain, but blind L1-L2 AND L1-L3 is a pain in the toot, could you not simply use L1-L3 and deal with the resulting stub?
This is however a question for your board house, capabilities vary widely.