Hi Abhijit,
Your journey sounds amazing and one of a kind. You can choose from following routes depending on your liking:
Mumbai>Indore>Udaipur>Pushker>Jaipur>Agra>Lucknow 2000Km drive.
Mumbai>Surat>Udaipur>Pushker>Jaipur>Agra>Kanpur>Lucknow 1800 Km drive.
You can take stops in the places you like. Use maximum day light and have plenty of sleep in the night. Road conditions are not predictable because of monsoon season so keep all the equipments you might need to fix a flat tyre or just make sure you are equipped to handle any breakdowns.
Here are some pet friendly hotels across india: https://goo.gl/m14KR5
And when you have finished your journey, don't forget to share your marvellous travel story with us here- https://www.tripoto.com/trips/create
Keep going!