Specific reaction parameter density functionals (SRPDFs), which can describe dissociative chemisorption reactions on metals to within chemical accuracy, have so far been based on exchange functionals within the generalized gradient approximation (GGA) and on GGA correlation functionals or van der Waals correlation functionals. These functionals are capable of describing the molecule metal surface interaction accurately, but they suffer from the general GGA problem that this can be done only at the cost of a rather poor description of the metal. Here, we show that it is possible also to construct SRP-DFs for H-2 dissociation on Cu(111) based on meta-GGA functionals, introducing three new functionals based on the "made-simple" (MS) concept. The exchange parts of the three functionals (MS-PBEl, MS-B86bl, and MS-RPBEI) are based on the expressions for the PBE, B86b, and RPBE exchange functionals. Quasi-classical trajectory (QCT) calculations performed with potential energy surfaces (PESs) obtained with the three MS functionals reproduce molecular beam experiments on H-2, D-2 + Cu(111) with chemical accuracy. Therefore, these three non-empirical functionals themselves are also capable of describing H-2 dissociation on Cu(111) with chemical accuracy. Similarly, QCT calculations performed on the MS-PBEl and MS-B86bl PESs reproduced molecular beam and associative desorption experiments on D-2, H-2 + Ag(111) more accurately than was possible with the SRP48 density functional for H-2 + Cu(111). Also, the three new MS functionals describe the Cu, Ag, Au, and Pt metals more accurately than the all-purpose Perdew-Burke-Ernzerhof (PBE) functional. The only disadvantage we noted of the new MS functionals is that, as found for the example of H2 + Cu(111), the reaction barrier height obtained by taking weighted averages of the MS-PBEI and MS-RPBEl functionals is tunable over a smaller range (9 kJ/mol) than possible with the standard GGA PBE and RPBE functionals (33 kJ/mol). As a result of this restricted tunability, it is not possible to construct an SRP-DF for H-2 Ag(111) on the basis of the three examined MS meta-GGA functionals.