suggested they do / suggested they would do?
Solution 1:
There'd be nothing ungrammatical about the version with would meet, however it would change the sense of suggest from definition 1 below to 2 or 3.
AHD
To offer for consideration or action; propose: suggest things for children to do; suggested that we take a walk.
To express or say indirectly: The police officer seemed to be suggesting that the death was not an accident.
To make evident indirectly; intimate or imply: a silence that suggested disapproval.
Compare:
Definition 3
She tilted her regal head in a gesture that suggested she would reserve judgment on whether or not this particular structure deserved the title. (Hearth stone, Lois Greiman)
Definition 1
While we were talking about his Opening Bat he told me about one of his serious collectors who’d sent him an email just after he’d received his copy saying “Is that it?”. Brian suggested he reserve judgement until he’d at least tried playing with it first, and several days later there was another email along saying “Ah, now I understand...” (blog - Puzzling Times)
The original sentence is a mandative construction (CaGEL p995). These can be either subjunctive mandatives (as in the example given), should mandatives, or covert mandatives.
They suggested he get there early. [subjunctive mandative]
They suggested he should get there early. [should mandative]
They suggested he gets there there early. [covert mandative]
On the acceptability of these, same source:
Clear cases of the covert construction are fairly rare, and indeed in AmE are of somewhat marginal acceptability. In AmE the subjunctive is strongly favoured over the should construction, while BrE shows the opposite preference.
Suggest has both a mandative and non-mandative interpretation - the addition of would into the sentence given changes it from a mandative to a non-mandative one.