"If I plant a double row, one row will be on the south side of the
trellis which is great. But the second row will be on the north side of
the trellis."

My 2 c's...

If I read this correctly, both rows will be climbing the same trellis? Unless this is a 3-dimensional trellis (such as an A-frame), then I assume both rows will be sharing the same area on the trellis - in which case the North row would not really be shaded by the South row, per se.

You could have an issue with crowding, though, depending upon the combined plant density of the two rows... so there may be little to no advantage of a double row, vice a single row. If planting only a single row would allow you to move the bean trellis further away from the cucumbers, that might be preferable. Provided that both the trellises are vertical & spaced 3' apart, there should be very little shadow... and the beans will most likely out-climb the cucumbers anyway.

You are correct. The north and south rows would be growing on the same vertical trellis. So I think I will do as per your recommendation and plant a single row. I will split my row 50/50 between Blue Lake and Purple Peacock pole beans. If I don't get enough with a single row this year, I will build an additional raised bed next year. Thanks for the advice!

Another suggestion: Consider sowing the beans in the two original rows that you planned, but sow them twice as far apart as you if planting a single row, but alternate the positioning. IOW, if you drew a line from the first plant in front to the next plant in back, you would be drawing a line like this VVVVVVVVV.

While you would end up with about the same NUMBER of plants as in a single row, those plants would have almost twice the ROOT SPACE and access to nutrients as a single row.