Some clarifications:

1. Only TK and A3 fly to AUH. LH used to until last year, but now discontinued the flight.

2. There is no bus service between Dubai International Airport and Abu Dhabi International Airport.. The main bus options between the two cities is as follows:

- Express bus service between Abu Dhabi city centre and two places in Dubai (one in Deira, the other one near the Marina).
- EK bus from Abu Dhabi city centre to Dubai International Airport. You're supposed to be flying EK to use this service - I'm not sure whether they check.
- EY bus from Abu Dhabi international airport to Dubai city - again, intended for EY passengers' use but I'm not sure whether they check.

If none of the bus options are suitable, then you would need to get a taxi or arrange a private transfer. Private transfers are typically fixed rate and range from 275-300AED oneway, whereas taxis are metered and seems to average around 250AED depending on the exact to/from locations.