There are epoxy repair systems for wood. It should work here too. Once it hardens, the epoxy is a tough, sandable, paintable medium that can de drilled. Make sure it keys properly into the hole, so that it does not just pull free under load. This may involve an undercut if necessary.
In fact, I just checked, and there are epoxies available specifically for masonry repair too, but the wood product should be sufficient here, and is easily found in most home stores.
An alternative might be to add a flat wood molding behind the entire length of rod. 3/8 of an inch thick should be sufficient. Paint to match the wall. Now anchor the curtain rod to the wood, using expanding drywall anchors, if necessary.
The "traditional" way to do it is, as you have suggested, to cut the bricks out by hand (and refit the cut bricks once you've done to give a tidy edge). It can be a time consuming job.
The more usual way to do it once the lintels are in place would be to use a disc cutter - usually a petrol driven one - (aka a rotary cutter or Stihl saw, after the common brand) with a diamond-tipped blade to cut through the bricks. This will make a lot of dust. You can get a water pack to reduce the dust (and cool the disc, reducing the wear on it), but it can still make for a messy operation, as the water will make the dust into a slurry that the disc will spray about.
You can hire the saws from pretty much any plant hire outlet, who will also hire the diamond tipped blades, charging you per milimetre (or half milimetre) of wear. That can be expensive though, so if you've a lot to do, it might be cheaper to buy the diamond-tipped disc from an independant supplier.
I've also seen what is basically a chainsaw with a diamond tipped chain for cutting through walls, etc, but I've not actually seen one used to comment as to how effective they are.
Don't forget that you should use eye, ear, and dust protection if you're doing work like this.
Best Answer
3M Scotch outdoor mounting tape may be something that will work for you. I've seen photos of the product used for attaching to brick, although the current collection of ad copy does not mention it. I've used the product to hold outdoor items and it's quite durable and strong.
The tape is a foam core with adhesive on both sides, allowing for irregularities in the surface. If it's necessary to remove for any reason, the foam has to be cut with a blade or some have used dental floss or similar cording to saw the foam.
Many locations have this product, Home Depot included. Image courtesy of Home Depot.
If easier removal is required, 3M Dual Lock is a velcro-type fastener. The product is not hook-and-loop but rather is a set of plastic panels with super strong adhesive, containing mushroom-head stalks that interconnect for a quite strong attachment. It's a bit harder to find and a bit more expensive than the foam tape.