I'm going to be alone here, but it does make sense, and it's why we vaxed our 2.5 year old.
I was due with a new baby right before pertussis season, and I knew it was a vax I would eventually get for her anyway, so I went ahead and started the doses before the baby came so by the time it was this time of year she'd have a few doses in her (the first few doses must be at least 4 weeks apart, and there needs to be at least 6 months between the 3rd and 4th dose)
Pertussis is so so scary in small babies. But I also hate giving such a reactive shot to such a small baby. So I feel it's a compromise to make sure those around baby are up to date.
YES, it is POSSIBLE to carry the bacteria into the home even if you are immune. If an immune child gets the germ on their hand (say from playing with an infected kid) and then doesn't wash hands before coming home, then touches baby with his germy hand before the germ dies away, then the baby could get it.
However, the germ is only going to live on his skin for a short period of time, is not going to multiply, and is easily taken care of by some hand washing.
Now, if the same child was NOT immune, plays with the infected neighbor kid, and comes down with it, he will actually be CONTAGIOUS with it. The germ will invade his body and multiply, and every time he coughs or sneezes the germs will fly out onto surfaces, his hands, the baby... And not just for a short time like the first scenario, but for a LONG time he'll be spreading the germs.
He will be spreading germs around MUCH better than that one little germ that he carried in on his dirty hand before the host-less germ died away.
So no, being immune does not COMPLETELY prevent transmission, but compared to how much an infected child transmits the disease, it is a GREAT reduction.
I wouldn't "count on" breastfeeding as a tried and true way to prevent pertussis. Yes breastfeeding is awesome and strengthens the immune system and passes antibodies, but it is not magic. Look at the history of pertussis, and you'll see that the period of most pertussis infant deaths is also the same time period were bottle feeding was practically non-existent.
I don't feel comfortable vaxing a small infant with dtap (my own personal gut feeling) but I feel fine after 2+ years of age.
So if your kids are going to eventually get it ANYWAY, why not do it now to create a little bit of a "wall" for your baby, rather than vaxing the baby. Maybe it's just me, but I just feel a whole lot more comfortable vaxing older kids than little babies. (especially since many times it requires fewer doses)
Also, yes it takes several shots to get the peak immunity, but that does not mean that the shots leading up to the last one are completely useless. Each shot (starting with the first one) increase immunity by a certain percentage. (and I believe you said your oldest has already received several, anyway?) So even if you only have time for one or 2 doses, it does lower the chances at least somewhat (and decreases the severity if caught)
You must not be in the US, as I don't recognize that brand name. So you may want to look up the catch up schedule for dosing intervals. the DTaP (US) can be given as close as 4 weeks together for the first 3 doses, but then no closer than 6 months between the 3rd and 4th and 4th and 5th (and if the 4th dose is after 4 yrs old, the 5th is not necessary)
Good luck, it's a hard decision.