Unlike finish and complete, the semantically related verbs accomplish, manage and achieve don't seem to select the ing-form complement at all regularly.
You can check by running individual raw Google and Google Ngram searches; 151 (but including duplicates) results for "accomplish mending", for instance, looks unconvincing, especially when the quality of some of the examples is taken into consideration.
Collins Cobuild does not include in its relevant section on verb patterns 'accomplish' as one of the verbs
(begin ... cease ... come ... commence ... continue ... discontinue ... finish ... get ... go ... (not) go ... keep ... quit ... resume ... start ... stop)
in the stop/start group taking the ing-form catenatively in 'phase structures' (but neither does it include 'complete').
These questions, in turn, raised questions about what was required of
students to accomplish reading in this classroom...
(Baker & Luke: Towards a Critical Sociology of Reading Pedagogy...)
doesn't sound totally unacceptable. Perhaps 'reading' is nearer the noun usage here, an object rather than part of a complex verb group, whereas 'climbing the mountain' forces us nearer to the verb (PP) usage.
I'd say the usage is unidiomatic / highly formal rather than ungrammatical.
The use of a to-infinitive after 'accomplish' is not acceptable.
'I accomplished the climbing of the mountain' gives a (slightly) less awkward-sounding direct object (but this still sounds ridiculously formal). As you imply, pronouns and (some) noun phrases are far more commonly used.
'I managed to climb the mountain' / 'I made it up [to the top of] the mountain' are far more conversational.