We use 'would' to express presumption or expectation in the past or present. The use of 'knew' puts the timing of the presumption in the past. Present presumption. 'I am sure he would be happy to see you'. 'That would be Steve at the door.' Would is always followed by the infinitive without 'to', so your sentence is correct without 'be'.

"I knew this would happen again" What is all this stuff about future in the past? Just learn the stuff and give yoru brain a rest.

"I knew this would happen again" I think the reason is that we use the base form of the verb; the infinitive without 'to'. 'I knew he would be happy." Here the verb is 'to be'; in the first example, the verb is 'to happen'.