How should one differentiate between a father and son with identical names and don't have suffixes?

Solution 1:

The first / second son could perhaps be Ben Jonson [junior] major and Ben Jonson [junior] minor. Here are some written examples.

That major / minor distinction is particularly associated with "public" (fee-paying, "elitist") schools for some people, but I see no reason why it wouldn't suit OP's context. The two sons wouldn't normally need to be further specified using junior (as opposed to Ben Jonson senior) because the father wouldn't be major or minor in the first place, so we'd know they must be the sons.

I do not endorse arbitrarily assigning a numeral (Ben Jonson II). It seems reasonable to me that someone like a schoolteacher might take it upon himself to assign identifiers like major / minor. But noting that Loudon Wainwright III's father chose to be known as Loudon Wainwright Junior (not Loudon Wainwright II), I think we should accept that certain forms more strongly require a person's explicit endorsement of the name by which they wish to be known (just as we normally allow that a name should be pronounced as close as possible to the way the bearer himself says it).