I can't give you a definitive answer, but I think I can show a line of reasoning that may provide a rough upper limit. By reading the wikipedia page on the future of earth, looking up normal mountain erosion rates and the corrosion tables for Titanium and stainless steel, I come closer and closer to a good answer.
As you mention, the sun will go into a red giant stage in approximately 7 billion years. If the capsule is still close enough to the surface, it would get melted.
Thanks to plate tectonics, no rock accessible to modern humans is known to be more than about 4.3 billion years old. (live science earth oldest rock) It is assumed that the forces of subduction have taken anything older into the mantle where everything gets melted down and reformed. Note that the majority of surface rock averages about 2-3 billion years old. No capsule that man can currently build (except perhaps from thick enough tungsten) could survive being taken down into the mantle. Even a Tungsten capsule would still allow the stainless steel plaque to melt.
The oldest mountain range is the Appalachians at roughly 1.2 billion years old. So even a capsule buried in the deepest heart of the newest mountain available will become exposed in less than that time frame.
Currently the deepest mine is the Mponeng gold mine at 3.9 KM deep. According to an article on Science Daily, mountains erode at the rate of .1 mm to more than 7 mm per year. (colder or newer/steeper mountains erode faster) Using 3 mm/year as an average, that means the capsule will become exposed in roughly 1 million, 300 thousand years. (3.9 km = 3900000 mm divided by 3mm = 1300000)
At one or more points, erosion combined with precipitation will allow the capsule to tumble down the remaining mountain side, likely damaging the structure, but I will not attempt to estimate that effect. As long as the capsule is only exposed to fresh water, corrosion is essentially nil, but under conditions of salt spray or acid rain, it corrodes at roughly 0.02 mm/yr. There seems to be no information about the specifications of the capsule the Scientologists chose to build, so I assume they used an off the shelf stock size that was still thin enough for a common welder to work with. That gives us an upper limit of 100 mm. I will further assume that they welded under ideal conditions of inert gas shielding, using Ti rod stock that matches the chemistry of the plate stock. Using the corrosion rates I found, that gives us an upper limit of 5000 years before the capsule is definitely breached, releasing the inert gas and allowing the environment in, at which point corrosion of the stainless steel plaque can be assumed to begin. The most corrosion resistant Stainless Steel chemistry is Titanium stabilized, which has almost the identical corrosion rate of Titanium alloys.
Assuming a 50 mm thick sheet of stainless steel, and assuming that the engraved words go no deeper than 1/2 of the total thickness, normal corrosion would render the text completely illegible in a little over 1200 years.
So, the longest I would assume the message contained in the capsule would survive would be 1,300,000 years until the capsule is exposed, 5000 years for the capsule to exclude the elements once exposed and 1200 years for the message to remain legible, for a total of 1,306,200 years.
Of course, there is no indication that the Scientologists used any serious earth-moving and mining equipment. There is no tailings or spoil visible in the images of the time capsule site, (Trementina Base) so it is highly unlikely to be buried all that deeply into the mountain. What text I can find describes tunnels "hundreds of feet" not the several thousand truly deep mining would involve. That drastically reduces the time required before the capsule(s) become exposed.