Yes you can neutralise it and precipitate out the copper as oxide or carbonate, but then you need to let it settle and separate the copper containing sludge for safe disposal, after mixing it with cement or similar to make it solid and reasonably inert.
IME, definitely do NOT neutralize it to precipitate out the solids. Some of the solids precipitate out, but what's left is a royal mess.
And I would certainly not mix it with cement. You will be left with copper salts and copper is worth money. And it is not some crazy poison. It is soluble in acidic water, in which form it is toxic to ocean invertebrates in high enough concentration, is all. What you got there isn't going to be a pimple on a pimple on the ass of a gnat.
I would definitely just let it evaporate in the sun. If you have a place where you can put it that is protected from rain, it won't take long.
I use a different etchant (cupric chloride), but I just leave my tank outdoors. If I don't use it for awhile, it dries up into dry green solid/crytals in very short order during the summer. What's left in 2L of used peroxide/HCl probably won't be all that much solid, I imagine.
Don't even throw it away. Keep it. When you have enough you can make a liter of cupric chloride by adding HCl and water to it.
In small amounts of it, we call this stuff hazardous waste. If you have enough of it, it's a refined chemical. It took work/heat to create this stuff, and returning it to the environment and letting entropy take its course is just simply wasteful.