Yes. It should be, at least for Lourdes, the ton royal in the 6th tone.

I have it here with ps. 116 in gabc, with rhythmic signs more or less how it’s actually sung. What is on Gregobase just takes the Solesmes version as written.

name:Laudate Dominum;
office-part:Psalmus;
mode:6;
%%
(c4)Lau(f)dá(gh)te(h) Dó(h)mi(h)num(h_) o(j_)mnes(ixi) gen(h)tes:(g.) * (:) lau(g)dá(g)te(g) e(g)um(g_) o(h)mnes(ixih) pó(g)pu(f)li.(f.) (::)
Quó(h)ni(h)am(h) con(h)fir(h)má(h)ta(h) est(h) su(h)per(h) nos(h_) mi(h)se(h)ri(h)cór(j_)di(ixi)a(i) e(h)jus:(g.) (:) et(g) vé(g)ri(g)tas(g) Dó(g)mi(g)ni(g_) ma(g)net(g) in(h) æ(ixih)tér(g_)num.(f.) (::) Gló(h)ri(h)a(h_) Pa(j_)tri,(ixi) et(i) Fí(h)li(g)o,(g.) (:) et(g) Spi(g)rí(g)tu(h)i(ixih) San(g.)cto.(f.) (::) Sic(h)ut(h) e(h)rat(h) in(h) prin(h)cí(h)pi(h)o,(h_) et(h) nunc(j_) et(ixi) sem(h)per,(g.) *(:) et(g) in(g) sǽ(g)cu(g)la(g_) sæ(g)cu(g)ló(h)rum.(ixih) A(g.)men.(f.) (::)

Yes. But it’s only done in chant for the rosary, and the ton royal is the same no matter the text attached to it.

And as I said…how it’s actually sung matters more than the transcriptions with fauxbourdon verses.