I'm not sure if OP's second "single sentence" version is technically a zeugma/syllepsis, but it's in that general area (the "deleted" main verb after have is kept, which doesn't match the "retained" form keep).
In practice native speakers do sometimes use such forms, but as a rule they're avoided. OP should use the first version, explicitly retaining the two different (past and present/future) verb forms.
Probably for purely stylistic reasons even OP's first version sounds just a little strange. Most native speakers would probably prefer to separate the two tenses even more explicitly...
I have kept this thing between us and will continue to do so
To illustrate exactly why the construction is problematic, consider...
"I have and will give you money"
...which could be interpreted as "I have given you money [in the past] and will do so again [in the future]" or "I have money, which I will give to you" (perhaps I never had money before, or never gave any to you).