How can I screw a small threaded rod in a bigger hole?
Sure. You tap (or drill out and tap) the larger hole for a convenient thread size, and use an M6 threaded insert with that thread size (from personal experience, I suggest not using the helical type unless you have to - if there is no space constraint, a larger-outside-thread solid insert holds up better.)

M6x1 with an outer of M10x1.5 seems to be a common solid insert size that should work with the constraints you've mentioned, in "drill out and tap" mode.

If you have a different mindset or set of tools, you can also weld the existing hole up, redrill and tap for M6 - most folks don't go there first, as getting a good fill on a small hole is tricky - but again, you can drill it out larger first if that is the problem, or drill it out larger and weld in a plug.