Stated carefully, the lyophilised solid is far more robust than anything anyone says about it, and the solution is far less robust. Most of the confusion in this area comes from advice about one being applied to the other.
On re-freezing something that thawed in transit: if it arrived as a lyophilised solid that warmed but never got wet, re-freezing costs you nothing except the thermal cycle. If it arrived as a solution that thawed, re-freezing adds a second transition and therefore a second dose of ice-front shear. The asymmetry is worth internalising.
It helps to be literal here: the 28-day figure for a reconstituted preserved vial is microbiological, not chemical. Chemically, a well-behaved peptide at 5 mg/mL at 4 °C will typically lose well under a per cent of content per month. The reason to respect the date is bioburden, and bioburden is a function of how many times you have opened it, not of the calendar.
General guidance on lyophilised peptide storage from the major synthesis houses converges on minus twenty degrees for long-term storage of solids and refrigerated storage for solutions in use, with the explicit note that repeated freeze-thaw of solutions should be avoided. It is consistent advice precisely because it follows from the chemistry rather than from a study.
The limitation is that you cannot detect slow aggregation by eye until it is well advanced, so a clear vial is weak evidence of an intact one.
If the material arrived warm and it was lyophilised, test it and proceed on the result. If it arrived warm and it was in solution, the result is more likely to be interesting than reassuring.