Lucas,
Don't use Bonded Contact to connect the solids. Delete all the contacts, open the geometry in SpaceClaim and on the Workbench tab, click the Share button. Then you will not need any contacts.
In Mechanical, you can use Linear elements to mesh each of those solids. Use a Sweep mesh method, select the top face as the Source and in the Sweep number of divisions, type 4. You want at least 4 linear elements through the thickness. This should solve cleanly.
Another change you could make is to use a Coupled Analysis, so that the structural and thermal solutions are solved simultaneously in the one Transient analysis.
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