Repeating blood tests for inclusion

Strict compliance with the approved protocol is the cornerstone of GCP. Adding things, that the protocol does not explicitly mention, will mean that this new extra-protocol process, will not have been formally approved by the Sponsor, the statistician, IRB/IEC and regulatory authorities. To change the protocol (unless to enact an Urgent Safety Measure) should be by an approved protocol amendment.
The approved protocol (approved by IRB/IEC and regulatory authority) should detail the acceptable parameters for re-screening & re-testing prospective participants. Repeating testing, until a prospective participant eventually complies with the selection criteria, may alter the assumptions made by the protocol. A statistician will have approved the protocol as it is and will not know about any unapproved changes.
Some tests may alter upon re-test. Some tests may require alternative forms to be used upon retest to remain valid. Some tests may require a period of time to elapse between retests and the protocol may have exacting requirements regarding timing of all tests.
With the protocol being “silent” on retesting, many research centres may correctly comply rigidly to the approved protocol. If you allow unapproved “additions” to the protocol, other centres may feel free to retest frequently. This might make a statistical difference between centres. This may also result in some centres having a large screen failure rate and others having a small one.
The approved protocol could also use Quality Tolerance Limits (QTL) on the selection criteria, stating what the limits were and what would happen is these were breached.
So probably a good idea to think deeply about rescreening and retesting and make sure that protocols cover this.