The conditions you describe is also involved in DNA isolation but with freezing ethanol and precipitation of DNA,  If the DNA can be pelleted there may still be a significant amount of your protein trapped in the interstitial spaces of the pellet with/without ethanol.  Is there RNA present as well or do you require RNase?  There is probably significant viscosity contribution from the lipid layers and peptidoglycan in combination with the detergent that might be addressed with dilution which might be OK for a large volume if the column is bind/release.  There's an "old school method" of DNA precipitation using protamine sulphate or  PEI but at the current viscosity you will likely have occlusion of the protein into the precipitate but perhpas prior dilution would help.   It's challenging to remove the DNA by cold centrifugation alone which I tried once at the 200 L scale when whole cells were accidentally solubilized in 6 M guanidine HCl.  If you're not allowed to generate aerosols then be also cautious of centrifugation where inconspicuous spills dry upon the rotor or chamber and can then become air born.  

You can try batch binding (without any column) on a shaker for overnight at 4°C, then make high speed centrifugation to pellet your resin, then wash the resin with any suitable buffer you want 2-3 times, then you can pack the resin into the column, and make elution of your protein with any suitable buffer.

To reduce viscosity due to RNA and DNA I typically use Benzonase from Sigma (http://www.sigmaaldrich.com/catalog/product/sigma/e1014?lang=en®ion=US). It is far more effective on DNA and RNA and works well at pH 7.0-8.0.  It is superior to DNase for the procedure that you are performing. In addition, you could always treat with Benzonase and then run your sample through a syringe with a small gauge needle on the end to sheer any existing polynucleotides to further reduce the viscosity. 

Was there sufficient Mg2+ in the sample to support the activity of the DNase?

I think the problem is that my lysis buffer is pH 12 which I need for my protein. The Benzonase will not be any effective at this pH I suppose.

Benzonase will retain about 90% of its activity at pH 9... but sure about pH 12. You may want to consider suspending the sample in a buffer at pH 8 to 9 and treat with the enzyme for 30 min. After that, bring the pH up to 12  for your downstream applications. Just a thought.

You might try to precipitate the DNA instead of sharing it before the affinity purification using either Protamine sulphate or PEI.

https://www.embl.de/pepcore/pepcore_services/protein_purification/extraction_clarification/removal_dna/

You can add glass beads (0.5 mm, aprox. 10 mL of beads for a 50 mL sample) and vortex the sample for a LONG time (10 min. approx). Passing the sample through a small gauge needle also works, but I'm not sure how compatible that is with your requirements (no aerosols).