Is there any way to use a display resolution higher than the max?

You could use a virtual desktop; this would allow you to scroll around to support more virtual resolution. But apart from that, no. Your LCD has a maximum resolution. By definition, that means it cannot support a higher resolution.

Both AMD and NVIDIA have Virtual Super Resolution (or Dynamic Super Resolution) configurations now in the latest drivers.

With AMD it should be pretty apparent in the Crimson driver.

Radeon Settings > Display > Virtual Super Resolution

As for NVIDIA, it's in the control panel:

Many monitors support many resolutions. If your monitor is capable of doing that resolution, then you might be able to get it to scale up a bit. But as far as exceeding a maximum resolution, the answer is usually no. The hardware generally lacks the ability to sync to a resolution that is not supported.