I know you asked about bins+camera phones, but I think your success will be limited. Expensive DSLR isn't absolutely necessary, though, I have gotten a lot of great pictures out of my cheaper-and-smaller Panasonic Lumix FZ200, which has a builtin 600mm non-DSLR lens -- getting a lot of optical zoom is critical.
I've done some digiscoping with a spotting scope plus this relatively cheap Gosky adapter from Amazon (this one would fit bins: https://www.amazon.com/Gosky-Universal-Smartphone-Telescope-Microscope/dp/B01D5W0WES/ref=sr_1_sc_3?ie=UTF8&qid=1486338210&sr=8-3-spell&keywords=gosk+y+spotting+scope). It's ok, but it takes a bit to setup -- you're not getting any warbler pics with this kind of setup, but it serves for ducks and other things that tend to be not moving.
What kind of birds are you hoping to take pictures of? And are you looking for pics-for-ID or pics-for-aesthetic-beauty?
I'd also be interested to hear about what people recommend. I was looking for one myself but I was not sure if the branded solutions from Celestron or Tele Vue, that I've seen, only work for their eyepieces or do they will they work on any eyepiece.
I saw this one Amazon and for the price I thought it would be good just to try it out.
I used this.
I used the following from this link on Amazon. I aligned the phone and eyepiece during the day so that I wouldn’t have to mess around in the dark
https://www.amazon.com/dp/B01D5W0WES/ref=cm_sw_r_cp_api_glt_i_BHPT3TVZQWV2W20SPDZR
I know Amazon sells such mounts. I am planning on buying one but have not done so yet.
The phone mount I've had for over a year now is this one I bought from Amazon (I use this with my Zhumell 8/12 inch dob with all my eyepieces, so it would definitely work with yours): https://www.amazon.com/gp/product/B01D5W0WES
It's made for larger phones in terms of width, but I know it can work with pretty much any size smartphone as I've used some of my friend's iPhones and such with it. With the phone mount, you can even start to capture some Deep Sky Objects with your phone. You will get nowhere close to the complexity that people can do with proper deep sky astrophotography setups, but with patience, good calibration with the phone and the eyepiece, and darkish skies, you can get decent results for what you have. I have a whole load of 1-2 second exposures of DSO's on my phone but just haven't uploaded them to my computer. I've captured many, many targets like the ghost of jupiter nebula, sculptor galaxy, ring and dumbbell nebula, virgo galaxies, globular clusters and open clusters, and so many more. They're all pretty much in monochrome as it's a single shot, but just being able to capture the basic structure of the deep sky object with just my phone camera is amazing, and I think everyone with a dobsonian should try to do these DSO's! Here's some results though that I've gotten with my 8/12 inch zhumell dob:
https://cdn.discordapp.com/attachments/516647701132345356/686760623404613647/20191118_192536.jpg (ring nebula)
https://www.reddit.com/r/telescopes/comments/bs7mb8/what_a_globular_cluster_messier_22_actually_looks/ (globular cluster)
https://www.reddit.com/r/astrophotography/comments/b0j270/waxing_moon_31219/ (first quarter moon)
https://www.reddit.com/r/astrophotography/comments/eyih92/sirius_with_its_white_dwarf_companion/ (Sirius with a white dwarf)
https://www.reddit.com/r/telescopes/comments/ddio34/is_it_a_bad_time_for_m31/f2l8xfp/ (andromeda galaxy and cigar / bodes galaxy)
https://www.reddit.com/r/astrophotography/comments/ba3d93/early_morning_planets_jupiter_and_saturn/ (jupiter and saturn a few months before opposition)
https://www.reddit.com/r/astrophotography/comments/efm00z/the_sun_with_solar_cycle_25_sunspots/ (the sun with my solar filter, same telescope)
As for image processing, it is definitely a little intimidating at first to learn how to turn a video of a planet or the moon (don't try deep sky objects with this method, turning those into pictures is a whole other ballpark) into a refined image, but if you look up some tutorials on youtube or ask around the subreddit, I'm sure you'll pick it up very quickly as I did over a year ago with said methods. The software will do most of the hard work and there isn't a lot you have to input besides a video, preferably at least a few seconds long and stable with your phone mount. If you would ever like help with anything related to telescopes feel free to ask here or send a PM to me! I was in your exact situation a little over a year ago and super eager to do photos with my dobsonian telescope, and have learned so much since then. Clear skies and nice picture again!
I used this adapter from Amazon: https://www.amazon.com/gp/product/B01D5W0WES/ref=ppx_yo_dt_b_asin_title_o07_s00?ie=UTF8&psc=1
With this.
I lined up my phone to the eyepiece while my scope was aimed at Polaris so I wouldn't start off chasing a moving target.