Ah, yes, I see your point.
Maybe it is important to point out that we are fitting different observations in the papers. In
1007.3725 you always fit the local [tex]\sigma_8[/tex], which weighs in on the allowed values for [tex]\Omega_{matter,in}[/tex], as you pointed out (private). In the fits where we included the matter power spectrum in
1007.3065, we also found much lower [tex]\Omega_{k,in}[/tex] and hence low [tex]H_{0,in}[/tex]. On the other hand, the profiles that led to a high [tex]H_{0,in}[/tex], change so significantly around the redshifts relevant for the large scale structure, that we didn't want to fit them with an effective FLRW. So I think the source of different [tex]\Omega_{k,in}[/tex] is not so much the prior, as it is the data that one fits.
About the global curvature being subdominant: yes, you're absolutely right, the main ingredient is [tex]\Omega_{k,in}[/tex]. But, the two values you quoted from our paper for [tex]\Omega_{k,in}[/tex], 0.98 and 0.93, are for models that do predict the same [tex]H_{0,in}[/tex]. The difference is the shape of profile and the global curvature..
It would be good to be able to calculate some LSS for the rapidly varying profiles too... Work to do!