Can a det. $k$-counter machine ($k \geq 2$) be simulated by a det. $k'$-counter machine using non-$\epsilon$-moves followed by $\epsilon$-moves?
No, that is not possible. There are languages accepted with $\varepsilon$-transitions that cannot be accepted if $\varepsilon$-transitions can only occur at the end (even if you
allow nondeterminism in exchange for restricted $\varepsilon$ usage).

Take, for example, the language $L=\{w\#w \mid w\in\{a,b\}^*\}$.
As a recursively enumerable language, it can be accepted by a
determinisic 2-counter machine. A simple counting argument shows
that this requires interspersed $\varepsilon$-transitions:

If $\varepsilon$-transitions occur only at the end, then
after reading a word $w\#$, the machine can only produce counter
values whose size is linear in $|w|$. In particular, there is
a polynomial $p$ so that after reading a word $w\#$, the machine
can be in at most $p(|w|)$ configurations. Thus, since there
are exponentially many words of a given length, the machine can
be tricked into accepting a word $w\#w'$ with $w'\ne w$.