The Table 2 in that study points to the chart of active vs non-active alloys to use. We think of lead being fairly non-reactive, but with seawater and CO2 attack, the lead portion of the solder is simply slowing going away. It is combining with Chlorine and leaches away.
Think about brazing the caps back on. I've not done a ton of brazing, but did do some using a copper alloy rod on copper fittings.
In your case, you want to inspect carefully the tube joints to the separator plates. These SHOULD be done with a higher melting point solder (or brazed) than the case and end caps. Otherwise, the heat required to pull off the end caps would disassemble the entire heat exchanger, tubes and all!
If they look good, I would certainly explore brazing the end spiders on. It might not be repairable in the future, but you could squeeze out quite a few more years out of them.