There is a celebrated paper by Ferrenberg et al, Phys Rev Lett, 69, 3382 (1992) who reported errors using the Wolff method for the Ising model, which they attributed to poor random number generators. However this pre-dated the development of the Mersenne twister, and it's not completely clear what recommendations came out of this, beyond "try several different RNGs": see, for instance, this preprint.
One of the most recent studies of the Ising model by the same group uses the Wolff method with the Mersenne twister RNG: see
Ferrenberg et al Phys Rev E, 97, 043301 (2018) also available as a preprint. So I would guess that they are happy with this RNG.
You might consult Deng et al Comp Phys Commun, 178, 401 (2008) who extol the virtues of "twisting and combining" random numbers, in their paper which is also targetted at the Wolff method. Unfortunately I can't find an open access version of this paper.
I think it's more likely that there is an error in your program. Or possibly you just need to simulate for much longer: you are, after all, simulating at around the critical temperature $k_BT_c/J\approx 4.5$. This is the wrong site to ask people for advice on debugging your program; for information on "how long to run" you should look at the literature (for example, the 2018 Ferrenberg paper cited above).
[EDIT following OP comment]
It is hard to judge whether your run parameters are satisfactory or not. The 2018 paper I mentioned is clearly "state of the art", using considerable computer power to get very precise results. But reasonable results could be obtained for this model, more than 25 years ago, using conventional updating schemes, see for example Ferrenberg and Landau Phys Rev B, 44, 5081 (1991) and, very roughly, your runs look to be of comparable length (measured in "complete lattice updates"). Your equilibration periods look somewhat short, but you seem to be systematically sweeping through the temperatures in very small steps, so maybe the equilibration period does not need to be too long. Even in that 2018 paper, they mention equilibration periods of $\sim 10^5$ Wolff steps, and my reading of their paper is that the actual equilibration was happening over a shorter period.
If you are not doing this already, you should at least be storing the histogram of energies at each temperature, and using histogram reweighting to compute results (average energy and heat capacity) at nearby temperatures, which will give you additional possibilities to check your results for self-consistency.
I cannot add any more to my answer: it would be guesswork, and nobody but you can check your program and your output in detail.