Let me give this a try:

It seems you are really asking about whether there are any outer derivations of $Lie(G')$ in $Lie(G)$. [EDIT] This does not, unfortunately amount to exactly the question of whether outer derivations exist, because they may not be realised inside the normaliser of $Lie(G')$ in $Lie(G)$ (a good example is to take $G'$ is a torus; then the normaliser in $\mathfrak{gl}(Lie(G'))$ is itself as $G'$ is then a maximal torus of $GL(Lie(G'))$). Of course there are no outer derivations in characteristic $0$ for the Lie algebras of semisimple groups. In positive characteristic the situation is rather different. For classical simple Lie algebras (i.e. ones coming from algebraic groups---there are plenty more simple Lie algebras in char $p$ like the Witt algebras $W(n;\underline{m})$), there are no outer derivations in very good characteristic because the Killing form is non-degenerate. (You could read Seligman Modular Lie Algebras, p112.) So if you can reduce to this case then I suppose you're alright. 

Now the Killing form is degenerate for $\mathfrak{psl}_{lp^r}$ (or indeed $\mathfrak{sl}_{lp^r}$) but you are ruling out these on the basis you are looking at the Lie algebra of an adjoint group and verily $\mathfrak{pgl}_{lp^r}$ has no outer derivations. However! There will possibly be an issue for $G_2$ in characteristic $2$ since the Lie algebra is isomorphic (qua Lie algebra) to $\mathfrak{psl}_4$ (amazing but true--e.g. count dimensions). Then this will have extra derivations and I would think this may give rise to a counterexample. Possibly also $F_4$ in characteristic $3$, $E_8$ in characteristic $5$, $\dots$ though I would take a small bet in favour of there are none except for $G_2$ in exceptional type. I would not be surprised to learn that there is a paper where someone has established exactly the outer derivations in all characteristics and for all isogeny types, but unfortunately I can't point you to one.

[EDIT2]: I just computed the $G_2<\mathfrak{gl}_{14}$, $p=2$ example in GAP and it's telling me that its normaliser is dimension $22$ and its centraliser is dimension $1$ ??? Leave this with me]

[EDIT3: Here's the code:

    gl:=FullMatrixLieAlgebra(GF(2),14);
    gap> g:=SimpleLieAlgebra("G",2,GF(2));
    <Lie algebra of dimension 14 over GF(2)>
    gap> mats:=List(Basis(g),i->AdjointMatrix(Basis(g),i));;
    gap> Bgl:=Basis(gl);
    CanonicalBasis( <Lie algebra over GF(2), with 27 generators> )
    gap> l:=[];              
    [  ]
    gap> for i in [1..14] do
    > temp:=0*Bgl[1];
    > for j in [1..14] do
    > for k in [1..14] do
    > temp:=temp+mats[i][j][k]*Bgl[(j-1)*14+k];
    > od;
    > od;
    > Append(l,[temp]);
    > od;
    gap> h:=Subalgebra(gl,l);
    <Lie algebra over GF(2), with 14 generators>
    gap> Dimension(h);
    14
    gap> LieSolvableRadical(h);                     
    <Lie algebra of dimension 0 over GF(2)>
    gap> LieNormaliser(gl,h);  
    <Lie algebra of dimension 22 over GF(2)>
    gap> LieCentralizer(gl,h);
    <Lie algebra of dimension 1 over GF(2)>]

So your proposed surjection breaks for $G_2$ in characteristic $2$. 

But verily I get your proposed surjection for $p=3$ (and $p=5$ which is a good prime but just to make sure). 

I even checked to make sure that $\mathfrak{pgl}_4$ has no outer derivations when $p=2$ and this does seem to be correct. I would have expected there to be just one extra outer derivation for $\mathfrak{psl}_4=Lie(G_2)$, but clearly these are not linearly equivalent in whatever appropriate sense. This must have a cohomological interpretation---probably some version of the $7$-dimensional module for $G_2$ is turning up.]

[EDIT4 According to GAP, the simply connected version of $F_4$ has no outer derivations in char 2 (where it is not simple) or 3 (where it is simple so adjoint and s.c. are isomorphic) and $E_6$ has no outer derivations in char 2 or 3 (again, it's simple so adj=sc), but I would guess it might take a day or two to complete $E_7$ and $E_8$. If you want, you can email me and I can do these for you.]