I can see your problem and still, not much light at the end of the tunnel.
1. To enable an engineer to check the work of another engineer, the checker should have at least the experience of the design engineer.
2. The Code and the local legislation protects the Buyer of the pressure vessel / exchanger, by forcing the equipment designer / fabricator to accept responsibility for the equipment supplied. In your boots, I would ask the Insurance company, if they are prepare to accept your personal design verification. The alternative is to hire a Third Party Agency for the design verification and fabrication inspection in your behalf. However, you have every right and obligation to check the input data the designer used in his design, for comformance with the purchasing documentation, including datasheets and your specifications.
3. Also, please ask yourself, would you feel confortable to accept and approve the design provided by the exchanger designer, if you could only produce some approximate calculations by hand, in order to check the supplier's software? Would your managers allow you to approve those calculations and drawings?
Best regards,
gr2vessels