We must deal with three questions: To what category of kli (utensil) does a car seat belong? If it is muktzeh, can you find a use for it that allows you to carry it? If it is forbidden to carry, may you push it along with permitted things (i.e., the baby) in the stroller?
Category – A car seat is a kli shemelachto l’issur (a kli whose main use is for forbidden activity). It is true that the seat is not directly involved in the forbidden activity (i.e., driving), unlike most cases of kli shemelachto l’issur, which are directly involved in a violation (e.g., pen, car, etc.). Yet, utensils which serve an otherwise permitted function within the framework of a violation are also kli shemelachto l’issur. Examples are phone books and car doors (regardless of activating lights).[1]
Possibility of moving such an object – One may move a kli shemelachto l’issur only for a permissible function or if it is in the way, but not to protect it.[2] You wish to protect the car seat from loss, which would be a problem. However, the Magen Avraham[3] permits one to contrive a use for a kli shemelachto l’issur in order to move it, even if his main intention is to protect it. The Mishna Berura[4] accepts the premise of a secondary intention, although perhaps only for an existing need, but the Machatzit HaShekel[5] and Yalkut Yosef[6] permit inventing a need. However, you must have a real plan to use the car seat after coming home, and some poskim require that for the use you devise, you must not be able to easily utilize a non-muktzeh utensil.[7] If you can be creative, you may be able to solve the problem.
Carrying with other things – Pushing a stroller with muktzeh in it is indirect moving, which is forbidden if done for the muktzeh item’s protection.[8] In our case, the stroller’s other contents, especially the baby, are more significant than the car seat, and you may push the stroller for their sake.[9] However, the gemara[10] says that if fruit and a stone are in a basket, one must shake out the stone if he can do so without causing damage. So must you remove the car seat or not put it in to start with? Most poskim rule that even if only the muktzeh will be damaged (or, in this case, lost), one need not shake it out.[11] Therefore, you should be able to leave the car seat in the stroller.
Yet, there is still a question if we consider that there are two ways to explain the permission to keep the muktzeh item in the basket when there is concern of possible damage. It might be because the act of moving muktzeh and non-muktzeh together is permissible unless one refuses the easy opportunity to remove the muktzeh. Alternatively, it could be considered moving muktzeh, though it is permitted because of the need. If the latter is true, then the need may be suspect here. After all, you want to create a situation where the car seat, which you may not move to protect, is in the stroller so that you have an excuse not to shake it off. Indeed, Tosafot and the Ritva[12] say that carrying the basket with the fruit and stone is carrying muktzeh and is permitted only because of need. (The Rambam[13] is ambiguous.[14]) We have not found a source that discusses purposely creating a “joint basket” of muktzeh and non-muktzeh before Shabbat, and we are inclined to consider it improper.
There are those who permit moving a kli shemelachto l’issur if one puts a non-muktzeh object on it.[15] Another possible, but not unanimous, idea for a fold-up car seat is to carry it in a backpack starting before Shabbat. This is because one may continue carrying a kli shemelachto l’issur when he started doing at a time or in a manner that it is permissible.[16]
Thus, we cannot rule out the grounds and means for leniency, especially in a case of need. You must consider which of the possible options, or a combination thereof, is most feasible and preferable.