If I break the computation into smaller pieces it works OK:

sage: K.<p,d,e,N> = FractionField(PolynomialRing(QQ,4,'pdeN'))
> sage: R.<x> = K[]
> sage: a = x^3-x^-3
> sage: b = x^5-x^-5
> sage: c = x^8-x^-8
> sage: X = p*a +d*b + e*c
> sage: H = R(x^8 * X)
> sage: f = H - N*b*c*x^16
> sage: f
> -N*x^29 + N*x^19 + e*x^16 + (d + N)*x^13 + p*x^11 - p*x^5 + (-d - N)*x^3 - 
> e
> sage: psi = cyclotomic_polynomial(30)
> sage: psi
> x^8 + x^7 - x^5 - x^4 - x^3 + x + 1
> sage: f.quo_rem(psi)[1]
> -d*x^7 + (p + N)*x^6 + (-p + d + N)*x^5 + (d - N)*x^4 + (-d - 2*N)*x^3 - 
> N*x^2 + (-p - d - e - N)*x - d - e
>

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