You not only need the adaptor plate for the Beseler 4x5 enlarger, but also the correct light mixing boxes. The boxes for the Beseler enlarger protrude about 2cm below the Ilford head so that the bottom of the box is just above the neg carrier. (By way of comparison, the boxes for the Durst L1200 are flush with the bottom of the Ilford head.)
Personally, I would only buy a working setup -- enlarger fitted with an Ilford head and with the matching mixing boxes (there are three: 4x5, 6x7 and 35mm).