Mazal tov! You seem to assume that Motza’ei Shabbat is the halachically preferable time. Let’s first check that assumption, and then we will be more equipped to deal with your specific question.
The Torah says that pidyon haben is to be done from the time the child is a month old.[1] The Talmud is replete with references to its being carried out after 30days. There are two basic ways to determine when this time has come. The Shach[2] says that pidyon haben should take place after an astronomical month (which is slightly more than 29½ days). The Magen Avraham[3] says it is performed on the 31st day of the child’s life, which, depending on the time of birth and the time of the pidyon haben, could be anywhere from just over 29 full days after birth to just under 31. Our clear minhag is to never do a pidyon haben before day 31. However, it is less clear whether this is sufficient or whether we must also ensure that an astronomical month is complete.
If we require both the 31st day and an astronomical month, then we can understand the rationale for the Shach’s[4] minhag not to do a pidyon haben at night. This is because it is common that the night of the 31st precedes the passing of an astronomical month.[5] Although most Ashkenazim follow the Shach’s minhag,[6] the reason behind it is less accepted. This is crucial for our case. The Sha’arei Teshuva[7] says that we do a pidyon haben during the day in order to better publicize the mitzva. This factor is less applicable today since people are more available to take part in such mitzva events at night than they are during the day. Other more esoteric reasons are proposed, as well.[8] One practical application that highlights the difference between the reasons for doing a pidyon haben during the day is when a pidyon haben is delayed until Motza’ei Shabbat. In this case, as the month is undoubtedly over, the first reason does not apply, and it becomes appropriate to do the pidyon haben at night.[9] According to the other explanations, however, a delayed pidyon haben is no different from a regular one.
Assuming that it is acceptable to do the pidyon haben on Motza’ei Shabbat, is it preferable, and, if so, how preferable? Although the prompt performance of a brit mila is more pressing than that of a pidyon haben, the Shulchan Aruch[10] does say to do a pidyon haben “miyad (right away) and not let the mitzva be delayed.” His source[11] seems to base the need for such diligence on the general recommendation not to delay mitzvot. While we do not want to underestimate this factor, general diligence does not get most of us up by sunrise to do the various mitzvot of the day. Additionally, the importance of doing a pidyon haben “right away” when it has already been delayed may be diminished.[12]
After learning some of the sources, let us put things in perspective. Any time from Motza’ei Shabbat till nightfall on Sunday is valid. It is unclear whether, all things being equal, it is preferable to have a delayed pidyon haben on Motza’ei Shabbat or the next day.[13] Either way, the halachic difference is not great. Note that it is prevalent nowadays to schedule a standard pidyon haben during the afternoon. Although there is a delay of a few hours, there is an advantage regarding the number of participants who celebrate the mitzva.[14] Therefore, feel free to do what works best for you, based on religious and/or personal considerations. As you have probably learned, the birth of a firstborn brings much elation but also a fair share of fatigue and even emotional stress. May you enjoy the mitzva of pidyon haben with peace of mind.