The house has a walk-out basement with 1 bed 1 bath and a full living area.We furnished the basement and rented it through Airbnb - the income fully covered our mortgage and utilities!

There is no wrong way to go about it, you can use OPM or "other people's money" or utilize a government downpayment resource or look into FHA loans to see if you qualify.For a first step get in contact with a mortgage broker to see where you and your financials stand as far as obtaining leverage on your money. 

I used the deposit to pay for the cleaning and anything left was applied towards unpaid rent or utilities.

The problem is the "Instagram society" we live in.

If you have a good rate then enjoy your low interest debt and utilize your money in a better investment.Forrest Faulconer 

- Rent for a 2 bedroom in Rocklin is around $2400 per month + utilities. - How much do property managers charge?

These can include -Additional savings or reserves that you, or you co-borrower may have-Proof of payments done on time for utilities and/or rent-A letter explaining how the applicant would be able to make mortgage paymentsFor reference on how to calculate your DTI, check previous posts!

Get with your lawyer, but maybe you could utilize special option period clauses & simultaneous closes to only sell your property if you found a suitable replacement?

Utilizing any/all of this information! 

Utilize FHA loan for first time home buyer which requires you to put only 3.5% as down-payment plus closing cost meaning with your share of 10 k you can buy a property worth under 150 k all by yourself( 5,250 down-payment + 3750 approx towards closing cost ) idle approach is to buy it only on  your name for house hacking and to involve partner for other investmentopportunity... hope this helps..