I think that you have two options ...
(1) Do the first stage of a SQUEAC investigation. This will give you a good idea of what is good and bad about your program, why defaulters are defaulting, &c. A lot of this can be done using routine monitoring data, examination of patient records, and some interviews with carers and other informants. You won't get a coverage figure but you will get information to help you improve the program and make a decent guess about what coverage might be. You could do this in a few days.
(2) Use SLEAC. This is a rapid survey method using a small sample size (e.g. n = 40 or smaller) that can classify coverage as (e.g.) poor, moderate, or good (using whatever thresholds you want to define these classes) in individual districts or clinic catchment areas. If the populations are small then the sample sizes may be reduced to (e.g.) n = 30. SLEAC surveys can be combined to produce a wide-area estimate. A single SLEAC survey might take 3 or 4 days.
You could do both ... use SLEAC to identify the best (or the worst) and then use some of the SQUEAC tools to investigate the best (or the worst) programs.
In your situation (i.e. you have strong suspicions that coverage is low) and I were really pushed for money I would do (1) and use the information provided to inform program reforms (i.e. identify and fix the problems identified) before spending time and money surveying to get an answer that you already suspect. I would do (1) a few times (three months apart) perhaps extending the work to include some SQUEAC small area surveys or SQUEAC small studies before surveying with SLEAC or doing a full SQUEAC survey.
I hope this helps. Let me know if you need more information.