I am reassured to be told that SSPX priests whom readers know do indeed follow the rules laid down by the Society. Furthermore, I would accept that the presence in the Sacristy of a picture of Papa Bergoglio, or the customary NOMEN SUMMI PONTIFICIS FRANCISCUS NOMEN EPISCOPI VINCENTIUS, would be a sufficient indication that all is well. I can only say that I wished to offend no-one but had heard that there were clergy who ... I will say no more.
Also, I know of a European city where the Bishop himself suddenly provided an EF Mass at exactly the same time as a very long-established SSPX Mass ... instead of providing people with a second EF Mass at a different time for their potential convenience. I cannot convince myself that this was an initiative likely to build up the People of God in a unity of sincere affection and trust.
I'm not a canonist; but I'll take the risk of answering two recent queries about attendance at SSPX chapels.
(1) Different vibes have come from different Roman authorities about this question at different times; my own pastoral and moral praxis has always been what the old manualists called Probabilist; so I would say that there is certainly sufficient doubt about this to make the liberty of doing it at least a probabilis opinio, which, accordingly, you are at liberty for sufficient reasons to follow. But ...
(2) Practically if there is a diocesan Traditional Mass reasonably handy, I think it is by far the best to go to that, not least so as to encourage the flourishing of Tradition within the mainstream. And ...
(3) Theologically I think one should be extremely careful about the rumoured risk that some SSPX clergy might not be entirely in line with the very proper course which has been steered by their Superior H E Bishop Fellay. It seems to me that a crucial question is whether a celebrant is really a Sedevacantist either quoad papam or quoad episcopum. How to find out? I would be inclined to ask "Father, I have a rather impertinent and personal question to ask you ... I hope you don't mind ... but it's important to me ... in the Canon of the Mass do you Name Pope Francis and Bishop X [the diocesan]?"
Actually, it's not personal or impertinent for you to ask this, because, although the Canon of the Mass is said secreto, it is a very important public Prayer of the Church, and the Naming of the Vicar of Christ and of the local Ordinary is a desperately significant marker of Communio (or the lack of it). So you really have every right to ask and to be told and to know ... but it is best to be diplomatic!
Remember: Communion with Rome is what the martyrs and confessors suffered for under Elizabeth Tudor! In 1558/9, the surviving primate, Archbishop Hethe of York, risked the royal wrath by delivering a courageous speech in the Lords in which he did not conceal his immense dislike of the Pope, Caraffa, whose malicious Westpolitik (he hated Spain, and England was allied with Spain) had led him intentionally to weaken and humiliate the English Church. But, good pope or bad pope, Hethe made clear the duty of being in communion with the See of S Peter. As did every single English diocesan bishop, large numbers of the Upper Clergy, and most senior academics at this University.
If a celebrant cannot say Yes, or declines to answer, then I wouldn't attend his Masses. I would, if necessary, go to a poorly celebrated Novus Ordo rather than to the Mass of a priest who, every time he says Mass, formally and deliberately sets himself before God out of Communion either with the Successor of S Peter or with the lawful bishop of the place. Remember that the Te igitur is not a prayer for the Pope or the Ordinary; it is a prayer expressing Communio with them ... or not. That is its function. This is important! Do you want to become, in effect, a non-Catholic?
If, happily, Father's answer is satisfactory, then your subsequent decisions should reflect prudent consideration of the good and/or harm that the various options open to you might have on yourself and your family and on the other worshiping Catholic communities nearby. No man (or family) is an island!
Never forget that there are loads of keen young diocesan priests out there just longing to discover enough well-instructed laity around for it to be possible for them to start an Old Mass! Have you ... delicately ... tactfully ... tried any of them out? Never forget that there are Catholic churches in which you may find the Melkite or Ukrainian or Ordinariate liturgies.
And if you attend the SSPX chapel, don't share in a lot of cheap schismatic chatter. It's not good for you and it's not good for those who hear you.