Does your version have a 3.3V supply rail, or 5.0V? (DFRobots says they produce both variants.) This value affects the range of viable design options available to you.
This strikes me as the most straightforward solution. OK, I understand you're not comfortable working with SMT devices at that scale. I'm not either. But I can be patient long enough - with the help of a toolmaker's microscope, or at least a very good desk magnifier - to get the job done on one or two boards.
Doesn't the official Arduino Pro Mini design incorporate a jumper (solder bridge) that completely disconnects the on-board regulator? Does your board have that feature? With the on-board regulator out of the circuit, you would be free to drive the Arduino Pro Mini from a regulator circuit using a device of your choosing. The venerable 78L05, or LP2950, are probably adequate for your application.
Or simply buy a version of the Arduino Pro Mini that can handle power input from a 9V battery. I believe there around half a dozen companies building the thing, and I know they don't all use that sorry excuse for a 3-terminal regulator IC.
Even prehistoric relics like the 1N400x series would do the job. But, the two junction drops would ALWAYS be in series - compromising your ability to extract the full electrochemical charge from the battery.
10V is the ABSOLUTE MAXIMUM rating - the "smoke point"; the voltage at which permanent damage is likely to occur. 8V is the guaranteed operating voltage - the maximum voltage for the device to meet its published performance specifications. What happens between 8V and 10V? We aren't told. Perhaps the device will drop out of regulation; perhaps totally disabled, or perhaps passing the full input voltage to the output; or maybe the device breaks into oscillation; or maybe it fails to meet some obscure specification that doesn't matter to 99% of applications. We don't know what the behavior will be, but as Clint Eastwood once asked, "Do ya feel lucky, punk?"
And keep in mind that some factory-fresh 9V batteries will measure 10V, or a bit more, across their terminals.