Turing Machine with additional finite memory of size $n$
I don't think that you can "embed" the extra memory into the alphabet and "carry" it with the head moving around without a $|Q|$ blow-up (blow-up comparable to the one of approach 1).

So a (weird!) approach to keep the size of $|Q|$ low, that can work is the following:


embed both the extra memory and $Q$ into the alphabet $\Gamma_M = \{H,h\} \times Q \times \Gamma \times  \Gamma^n$
use only ONE cell (marked with embedded symbol $H$) to represent the current state of $M_{mem}$ + symbol under its head + the content of the extra-memory 
use some additional states to SHIFT the tape around the cell $H$


Very informally you have:

                  q1
                  S
....0 1  1 1 0 0 [0] 1 1 0 0 0 ...
    h h  h h h h  H  h h h h h 
                  ^
                  q_sim


In this condition, the (standard) Turing machine $M$ is on state $q_{sim}$ and is on symbol $\langle H,q_1,0,S \rangle$ that represents: head of $M_{mem}$ is on state $q1$, reading a $0$ and the content of the extra memory is $S$

Now if $M_{mem}$ writes a 1, writes $S'$ on the extra memory, moves $L$eft and enters $q_2$; the standard $M$ on state $q_{sim}$ writes $\langle H,q_2,1,S' \rangle$ and enters a sequence of states $q_{r_1},...,q_{r_k}$ that shift the whole tape one cell to the right and returns back on the cell marked with $H$ and state $q_{sim}$; simulating in this way the movement of the $M_{mem}$'s head: 

                  q2
                  S'
....  0 1  1 1 0 [0] 1 1 1 0 0 0 ... &lt;&lt; tape shifted right
      h h  h h h  H  h h h h h h 
                  ^
                  q_sim


Note that the standard $M$ simulates $M_{mem}$ transitions using only a single state $q_{sim}$ (plus two extra-states for accept/reject).

Also note that the symbol corresponding to the extra memory $S'$ is NOT MOVED, so the total number of states of $Q_M$ required to shift the "underlying" tape (simulating the move of the head of $M_{mem}$) depends only on $\Gamma$, not on $n$:

$\Gamma_M = \{H, h\} \times Q \times \Gamma \times \Gamma^n $

$\delta_M = Q_M \times \Gamma_M \to Q_M \times \Gamma_M \times \{L,R\}$

and $|Q_M| = k |\Gamma|+c$ 

You can also choose to embed the states of $M_{mem}$ into $Q_M$: in this case the alphabet is: $\Gamma_M = \{H, h\} \times \Gamma \times \Gamma^n $, but you must use more states: $|Q_M| = k |Q| |\Gamma|+c$

The above construction can also be generalized to standard Turing machines and (very informally) it roughly represents a switch from:

$\delta : Q \times \Gamma \to Q \times \Gamma \times \{L,R\}$ to

$\delta : \Gamma \times Q \to \Gamma \times Q \times \{L,R\}$