The phrases quantitative and qualitative methods/analyses/data get used differently by different people (and differently applied to each of those nouns). Those, Bryman statement is probably based on how the distinction is defined earlier in that work (also, the word "survey" is vague). Similarly, sometime philosophical school labels get used differently. If you have a particular and well-thought-through philosophy of science that you are happy with, I would not change your entire way of thinking because of collecting data in one way versus another.

What are your research questions? It is often easier to decide how to approach data and methods based on these, rather than trying to fit into labels.

Does this make sense?

Before answering your question, there is a threshold question. Which form of critical realism are you talking about? If you are referring to Campbellian Critical  Realism, then you are in essence talking about positivism. If you are talking about Bhaskarian Critical Realism (BCR) then there is more work to do. For the rest of my response, I'll be talking about BCR.

BCR holds to a structured and layered ontology. In this ontology, real things make every thing that comes to pass, pass. However we only have access to the empirical realm which means that our knowledge of the real is partial and biased. 

What we know as quantitative and qualitative analysis only deals with the empirical realm and does not penetrate to the real. So none of those research methods can provide explanations or proofs of explanations. To find explanations, we transcend the empirical facts and ask the question "What must reality be like for the events to occur as they did?" In this transcendental analysis, we develop models of the real things that might have caused the events to occur and select the one with the greatest explanatory power.

Now to you question. When we get quantitative or qualitative results, recognizing that they only deal with the empirical realm we recognize them as doing work to carefully define the phenomenon and where it occurs over what conditions. Then you explain how how that phenomenon occurred by creating a model of the real things and how they interact with the environment. 

So in your research you use your quantitative and qualitative methods to define the phenomenon and then use this definition as the basis on which to form your development of possible explanations. You could setup a series of contrastive demi-regularities (where it does and does not occur) and then use that as the basis to create the explanations (Lawson, 1997).

So to directly answer your question, BCR is compatible with quantitative research methods just as long as you do not use them to generate or test explanations. 

I'm not clear why you are getting into philosophical issues when your primary concern seems to be whether it would be a mixed methods study if you combined the data from standard survey items with open-ended questions from the same survey.

In this regard, note that what Btyman said was "It might be argued that asking a small number of open questions in the course of a structured interview does not really provide an instance of [mixed methods research]" -- not that he himself felt this way. In addition he was taking about which studies to include in a search, and he did indeed include studies such as yours in his search.

So rather than taking Bryman as definitively rejecting what you did, you should provide a clear justification for how your qualitative data will contribute to the overall results from your study. In particular, if it helps you produce results that would not have been possible with a "mono-methods" study (the survey) then I would certainly consider it to be mixed methods.

I  argue that quantitative practices can be informed by critical realism in this chapter

https://www.researchgate.net/publication/256801634_The_practice_of_quantitative_methods

Chapter The practice of quantitative methods

"...administered through a survey carried out face-to-face";  "...data obtained through survey"

These phrases indicate the issue that I have been dealing with for the past 10-15 years in both my teaching research and working on research projects: the word SURVEY.

Survey is a research DESIGN or METHOD-- and originally referred to large scale research/samples before it was used with/in small studies. Although, the word, survey in general, can be used beyond research. (e..g. survey land, survey of the land), this is an example of why in research, we must be selective and specific in our choice of words and their usage and meaning.   When and how the word came to mean (which it does NOT) a "questionnaire" I have no idea.  So, when you say "administered through a survey", you really mean administered through a questionnaire, ..."carried out face-to-face with some open-ended questions." 

When someone says," I conduced a survey of, e.g. college students, nurses, executives, etc., it used to be understood that the subjects were studied using questionnaires (albeit long ago).  As a method, the researcher, now has to explain what kind of survey and how it was conducted; never the less, it still means a DESIGN or METHOD. I cringe when one my students or research coworkers, says, "we now have 35 of 200 SURVEYS completed", when they are clearly referring to the questionnaires.

Because survey research originated with large scale studies, many purists would argue that it is and can only be quantitative, (as would I, but must acquiesce to changes and adaptations). Questionnaires (these are tools/instruments) are, of course, used in qualitative research. Perhaps this is where some  of the confusion lies. "During my data collection I used an instrument (questionnaire) administered through a survey carried out face-to-face with some open-ended questions that require a narrative answers."  Sort this out, for research's sake:

First, what is your design-- mixed methods: which is the qualitative method(s) and which is the quantitative method(s)? Then, what are your instrument(s?  Then, the process of the method-- No that is not redundant, because in research, you have to detail precisely and step-by-step, what you will do/ or did-- in the order that you did it. I always had to remind my students, that their/your research has to be written in such a way as to be replicated, i.e conducted again by another researcher.

I recommend that you look at any articles or books dealing with surveys as qualitative methods, such as this: http://www.qualitative-research.net/index.php/fqs/article/view/1450/2946

Good luck, Usman.