All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
There is a difference of opinion about the hair of the dog whether or not it is impure. The view adopted by Shaykh Ibn Taymiyyah is that the hair of the dog is pure, and this view may be more suitable for you.
Therefore, it is not an obligation to purify anything on which the hair of dogs is found.
In addition, you do not have to purify anything you doubt that the saliva of dogs may have made it impure, because in principle it is pure (until proven impure). You should act according to this principle until the contrary is proven with certainty.
However, if you are certain that something is stained with the saliva of those dogs, then according to the view of the majority of the scholars, you have to wash that place only seven times, one of which with soil. Soap and the like serves instead of soil in washing.
Some scholars, like Ibn Al-Qayyim hold the view that washing seven times and using soil is only obligatory in washing utensils.
Therefore, it is enough to pour water on this impure place until you know or predominantly think that the impurity disappeared. Hence, there is no harm on you to act according to this view due to the hardship.
Of course, all of this applies if you are certain that there is impurity. Nevertheless, if you only doubt it (to be impure), then you are not required to do anything.
The bucket that you used to purify that car is pure. As for the floor of the car, if it is a glazed item that does not absorb impurity, then purifying it is achieved by wiping it until the very impurity disappears; this is the opinion of many scholars, and it is also the view chosen by Shaykh Ibn Taymiyyah .
Therefore, you may act according to this view that as it is easier for you and will relief from hardship and harm.
For more benefit, please refer to Fataawa 358072 and 228397.
Allah knows best.