Inverting buffer with op-amps

No, there is no way to make an inverting buffer with just an op-amp that does not depend on the resistor values. You can get resistors with very fine accuracy and stability (at an equally impressive price) or you can get networks with matched (in value and in temperature coefficient) where the absolute accuracy may not be so impressive but the ratio is tightly controlled.

There is a way to invert a signal without accurate resistors- the so-called flying capacitor method, but it's fairly complex and resistors are a better solution for most situations down to ppm level accuracy.

One possibility. They used to make a few specialized op-amps that had differential outputs, a positive and a negative, likely for driving differential line pairs. I've never used one and I don't recall the part numbers. But I assume if you were to connect the positive output up as a voltage follower the negative output would be equally negative.

Note that even a voltage follower op-amp arrangement is not perfect. There are internal gain spec's and small offsets, while small, can results in outputs that are not perfect one to one of the inputs.

Adding to Nedd's answer, the precision unity gain differential amplifier ICs you are looking for are of the type INA105, DRV134, THAT1240 etc. There are several qualities, makes and manufacturers and of course a varying price range. The circuit in question would be the one below, from the INA105 datasheet.