I haven't observed appreciable reciprocity failure with paper negatives in exposures many minutes long.
A convenient method of metering for pinhole, if you have a handheld meter, is to set your meter to your working EI (exposure index); I rate my grade 2 paper at EI=3. Then meter the scene, and refer to the f/stop of your pinhole divided by 10. That is, if it's an F300 pinhole, then look for the recommended exposure time opposite F30 on the meter scale (you do this because most light meters don't read above about F128). Then simply multiply this recommended time by 10, to yield your actual exposure time.
As a hypothetical example for an f/300 camera, I set my meter's EI to 3, meter the scene, then refer to the recommended time opposite f/30, which may read, say, 5 seconds. Simply multiply this time by 10, resulting in a recommended exposure time of 50 seconds.
Since this exposure time is well in the range of the times one may encounter in projection enlargement in the darkroom using photo paper, don't worry about reciprocity failure.
If your negative comes out appreciable underexposed using this method, it means that your working Exposure Index is too high, or you aren't developing long enough, or both. You can therefore use this method to dial in your working EI for your paper and processing methods.
I've found that I typically develop my paper negatives for 3-4 minutes, rather than the 1-2 minutes I may use for developing prints, but do so using a more dilute solution, so I can pull it at the right time.
PS: Regarding your posted image, is the fogging coming from the camera, the makeshift darkroom, or both?