I cannot find any published research that specifically considers using 2 dogs vs. more than 2 dogs to hunt foxes. However, there is a substantial literature on the effect of fox hunting on fox populations in general. All the studies I found indicate that fox hunting with hounds generally has little to no effect on fox populations.
The reasons for the ineffectiveness of hunting with dogs are well summarized by this study, which models the breeding and dispersion of fox populations, when subjected to different population control mechanisms. Among the study's (theoretical) conclusions is "We conclude that effective control of populations at
landscape scales (e.g., 1,600 km2
) is not feasible or practical unless immigration from outside populations is low or can be controlled. These
results can be used to inform policy on the management of United Kingdom (UK) fox populations and contribute to the ongoing debate on
hunting with hounds as practiced in the UK". Reading the rest of the study, we find that hunting with hounds is ineffective because:
Hunting generally takes place after the fox breeding season, when foxes are most numerous (and, presumably, most annoying to farmers and the like).
Foxes breed extremely efficiently. A large fraction (about 40%; presumably mostly juveniles) must leave their home environment soon after the breeding season because of a lack of carrying capacity (food for all the foxes). Most of these fail to establish themselves elsewhere, and die of natural causes.
Hunting with dogs typically only kills foxes that would have died anyway. The more foxes that are killed, the easier it is for the remainder to find food.
Winter hunting (which does not use dogs), is apparently somewhat more reliable, since it kills foxes before they can breed.
Although the findings of the first study are based on a theoretical model, they appear to be well supported by the available empirical evidence. This 2002 paper (apparently originally published in Nature) claims that during the fox hunting ban produced by an outbreak of Foot-In-Mouth disease, fox populations did not increase overall. The study's authors state "Our results argue against suggestions that fox populations would increase markedly in the event of a permanent ban on hunting."
Another paper, published in Science, interviewed a random sample of several hundred farmers, gamekeepers, foresters, and related professions, from across Britain. A large majority of this group (~2/3) believed that a ban on hunting with hounds would have no impact on their ability to control the fox population with lethal methods.