I have a proof of the integrality of $\Delta_3$ using Dwork's lemma (which tells when a given vector $\left(v_1,v_2,v_3,...\right)\in A^{\left\lbrace 1,2,3,...\right\rbrace}$ over some commutative ring $A$ is the vector of ghost components of a big Witt vector -- note that this is equivalent to the existence of a ring homomorphism $f:\mathbf{Symm}_{\mathbb Z} \to A$ which sends each power sum $p_n$ to $v_n$) and some binomial coefficient congruences (I am eventually going to write this up, though I cannot give a good upper bound on the "eventually").
Meanwhile this stuff doesn't seem so new. On page 14 of Andrew Baker, Birgit Richter, Quasisymmetric functions from a topological point of view, arXiv:math/0605743v4, I see a coproduct $\psi_{\otimes}$ which is exactly mine if I get it correctly that $q_0$ is an arbitrary integer. They seem to have a proof based on topology. They don't seem to notice the need for renormalization, though.
On the other hand, on the level of "algebraic rings" (commutative algebraic groups with an additional algebraic monoid structure that distributes over the algebraic group structure), the Hopf algebra $\mathbf{Symm}_{\mathbb Z}$ equipped with the comultiplication $\Delta_3$ seems to more-or-less represent the ring $\widetilde{A}$ defined in §3 of Berthelot, Grothendieck, Illusie, SGA 6, exposé 1. I am sorry for the weasel words, but I don't have the time to make more concrete assertions these days.
EDIT: Two (or three, depending on how you count) proofs of the integrality of $\Delta_3$ are now in Vic Reiner's and my notes on Hopf algebras in Combinatorics (the version with solutions). See Exercise 2.75(f) and Exercise 2.80(e). (The numbering is volatile, but the first exercise has the words "Define a $\mathbb Q$-algebra homomorphism", and the second exercise talks about "new solutions to parts (b), (c), (d), (e) and (f)".)