Dwork-type towers for the Catalan–Larcombe–French numbers and a transfer principle for Apéry-like numbers

Let P̄_n = Σ_k C(n,k) C(2k,k) C(2n−2k,n−k) (OEIS A081085), so that 2^n P̄_n are the Catalan–Larcombe–French numbers; P̄_n is the sporadic Apéry-like sequence E in Zagier's classification. In his list of open conjectures on congruences (2019), Z.-W. Sun conjectured two congruences modulo p (Conjecture 62) for the truncated sums of P̄_k/4^k and P̄_k/8^k, that is, at the two singular points x = 1/4 and x = 1/8 of the recurrence of P̄_n, where the series Σ P̄_k x^k diverges. The case n = 1 was proved by G.-S. Mao (J. Number Theory, 2017). We prove the integrality part of both congruences for every odd prime p and every n ≥ 1: the partial sums S(N) = Σ_{k<N} P̄_k/4^k and S_8(N) = Σ_{k