Personally, I think strangles should be a notifiable disease. At present there is no obligation placed on the owner(s) of infected animals to bring it to anyone's attention, so there is potential for horses to be moved around without impediment. If the disease were notifiable, there would be restrictions imposed on infected yards, and public awareness would be raised as to symptoms and treatment.
Unfortunately, there are many bacterial infections in horses which can be confused with strangles. Some of those are very similar in that they cause runny nose and swollen lymph nodes so, although strangles may be suspected, it is only identifiable in the early stages by clinical inspection of a bacterial culture. This may lead to strangles cases being mis-diagnosed by lay persons and, without the attention of a vet, going undetected and unreported, which may possibly result in infected horses coming into contact with others.
Nursing of strangles cases requires strict attention to equine and personal hygeine as it is a highly infectious disease.
On that basis, given what you say about horses going out from the yard to compete and, given that some horses can be carriers of strangles without showing symptoms, and as your shetland is only transient at the yard, it seems unlikely that testing him would be strictly necessary or sensible. Rather, I would say a period of quarantine on arrival on the yard would do more to assess the new arrival and help prevent introduction of the disease.
On the subject of introducing the two, it may be better to bring both animals to the new place at the same time and introduce them there. That way neither will have the upper hand in terms of territorial awareness/advantage, and there should be plenty to take their attention other than hierarchy issues. Without the complications of other horses, they are likely to form a mutual bond more quickly. However, that would only be sensible if you were convinced that the new pony did not present a strangles threat.