Poskim[1] discuss the case of a woman who had been pregnant from another relationship before her marriage. Her husband did not know this and assumed that their firstborn son required a pidyon haben. Could the woman allow her husband to do a pidyon haben for their son, including making two berachot l’vatala, to avoid embarrassment and possible repercussions to their marriage? The consensus is that considerations of k’vod haberiyot (preserving human dignity)[2] allow her to not reveal the truth.
One factor in the leniency is that most Rishonim maintain that a standard beracha l’vatala is only a Rabbinic violation,[3] and k’vod haberiyot overrides Rabbinic laws.[4] (After all, the content of the beracha, which says that HaShem commanded us [plural] in the mitzva of pidyon haben, is always a true and positive statement.) Furthermore, the wife was passive in this case; she did not stop her husband from making a mistake. The Rosh[5] says that under such circumstances, k’vod haberiyot supersedes even a Torah law.
The case you present, in contrast, is worse in several ways. First, the father knowingly is making a non-mandated beracha. It is true that he can technically get around that problem by making the Shehecheyanu over new clothing and by mumbling the beginning of the main beracha on the pidyon haben, while not uttering HaShem’s Names. However, there is another issue: Those assembled will answer amen to a beracha that is invalid, which is forbidden.[6] This ostensibly violates the Torah-level prohibition of lifnei iver lo titen michshol (facilitating a person’s act of sin), certainly when one consciously causes others to do so unwittingly.[7] One cannot invoke the ruling of the aforementioned Rosh in this case because the father is actively presenting, rather than passively not stopping, a “beracha” to which it is forbidden to respond.
Nevertheless, it should still be permitted. Consider the following. It is forbidden to daven when one has to use the facilities, and if the need is acute, his tefillot and berachot are invalid.[8] Still, the Bi’ur Halacha[9] states that a chazan in that condition, who will be very embarrassed to walk out in the middle, may continue leading the tefilla. In that case, the chazan knowingly improperly makes berachot to which people will answer amen, and yet it is permitted due to k’vod haberiyot. This seems to be a pertinent precedent for our case. The explanation apparently is that just as the chazan is allowed to violate a Rabbinic law for the sake of k’vod haberiyot, he is allowed to make others violate the Rabbinic law for that purpose. Since it is permitted under the circumstances, lifnei iver does not pertain. Similarly, in our case, if in the face of a severe concern for k’vod haberiyot, it would be permitted for the father to actually make a beracha l’vatala, it would likewise be permitted to cause people to answer an improper amen in unintentional violation of a Rabbinic prohibition.
In spite of the above, it is worthwhile to encourage people not to make a fake pidyon haben when not extenuatingly necessary. The fact that a few close friends and relatives find out about a miscarriage is not always as embarrassing as it seems to some or at first. It is also possible for a couple in this situation to say that the birth was a caesarean delivery, which exempts from a pidyon haben,[10] or that the delivery was with forceps, which calls for a pidyon without berachot.[11] However, every case is unique, as is every person, and you should therefore look into and ask for halachic advice regarding each case if and when it arises.