If one takes a very purist stance then it would be very difficult to integrate standard CT with mindfulness/acceptance based therapy. This because CT essentially does agree (collude?) with the client in that the change in cognitive content is supposed to facilitate/cause change in emotional experience and corresponding behaviour. The implicit message is: it would be really bad if you felt like that but you can change your experience through changing the content of your thinking. Mindfulness/acceptance based therapy on the other hand gives the message: we're compassionate about you having this experience AND part of your problem is that you don't want to have that experience AND your feelings and thoughts are just that, namely thoughts and feelings you don't have to act upon. Mindfulness/acceptance is definitely not just another "little technique" as it is presented sometimes by second wave therapists.

In practical terms I think we need to meet our clients were they are in regards to their culture, education and upbringing. For most people living in a "western culure" this means a high premium is put on analytical thinking and concrete problems solving, which both are discouraged in mindfulness/acceptance approaches (at leas to some degree).
However, even stock standard CT techniques such thought record and doing a "5 part model" do contain elements of non-judgemental observation of external and internal stimuli. Therefore it seems very possible, or maybe even necessary for some clients, to "prepare the ground for mindfulness/acceptance strategies through conventional Ct techniques which already facilitate some "decentering/development of an observer self".
(See also research on efficacy of MBCT between clients with only 1 or 2 depressive episodes compared to those with more than 2!).

It also seems very useful to use CT techniques to elicit the clients intermediate or core beliefs/schemas which facilitate ruminative processes through comparing and judging one's external and internal environment against some sort of ideal standard and expectation (again see MBCT model). Then, of course the decision will have to be made as to promote the "restructuring" of these beliefs (and derived automatic thoughts) or the non-reactive observation of them.
Having a reasonably coherent model of how mindfulness/acceptance intervenes in the otherwise incessant process of comparing and judging and behavioural responding to internal and external stimuli will be helpful in the therapy process of shifting the clients awareness from the content to the process of cognition (including processing physiological and emotional experiences).
From my own experience, it seems possible to integrate CT with mindfulness/acceptance strategies in a staggered fashion whereby cognitive restructuring prepares the ground for genuine mindfulness/acceptance work. It also depends on the severity of the client's presentation and the degree of the "irrationality" of their thinking. The more the client's distress stems from "dysfunctional" cognitions the more appropriate is CT. The more distress is attributable to realistic cognitions (about real life problems such as death of partner, own illness, financial situation etc.) paired with non-acceptance of these inevitable (in the moment) stressors the more mindfulness and acceptance based work appears to be the way to go.

These are exciting times and we may see a development that rivals that of the advent of BT or CT.