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@moth
transverse fibres would resist the propagation of longitudinal cracks, but they are not the problem. Transverse cracks are.
If some longitudinal fibres break you don't want their share of the tension transferred to their neighbours in the immediate vicinity of the break: that would be concentrating the stress in those few neighbours, and transverse fibres certainly don't act against this. The sharp edge created as the transverse fibres separate would focus the stress even if they aren't directly carrying the tension sideways.
What you do want to do is gradually redistribute tension from the broken fibres among all the rest, so that along the spoke from the break the broken fibre is still carrying a full share of the stress and can even take its share if other fibres are broken elsewhere.
I can see you could argue a case for shallow angle diagonal fibres (something like a traditional natural fibre rope), but maybe you can achieve the cross-transfer just with the resin bonding between the fibres. At the micro-structure level individual carbon fibres are formed from many shortish filaments tangled together and only gripping each other with weak intermolecular forces.