How can I input a custom script into the WMS request URL?

Add the EVALSCRIPT parameter to your OGC request URL with the value of the base64 encoded script, as such: evalscript=cmV0dXJuIFtCMDQqMi41LEIwMyoyLjUsQjAyKjIuNV0%3D

More on custom scripts described here.

A working WMS request from EO Browser, using a simple custom evaluation script: https://apps.sentinel-hub.com/eo-browser/?zoom=10&lat=41.6429&lng=12.8862&themeId=DEFAULT-THEME&visualizationUrl=U2FsdGVkX1%2FEsuqLi%2Bg9qu%2F0sihYyvSWA9Vkx%2FWiwVwE4Qceogyb7GRdjhPMAkgAUb%2FSqI%2BswsF4h78zwZKmhK%2BEm0G3AAI6Q7YCYJS6tvS9PdQVkl%2FVqgKUKTR1p8QF&evalscript=cmV0dXJuIFtCMDgqMi41LEIwNCoyLjUsQjAzKjIuNV07&datasetId=S2L1C&fromTime=2020-05-15T00%3A00%3A00.000Z&toTime=2020-05-15T23%3A59%3A59.999Z#custom-script

Note that the LAYERS parameter should be set as one of the existing layers in the configuration when using a PRESET=CUSTOM with a custom evalscript.