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This is not an answer, but a possible clue that it is a problem of ARB and not the SageMath wrapper.

When I run this code in my IDE:

#include "flint/arb.h"

int main(int argc, char **argv)
{
    arb_t a;
    arb_init(a);

    slong prec;
    prec = 256;

    arb_set_d(a, 1.1);
    arb_printd(a, 40);
    printf("  WRONG \n");

    arb_set_str(a, "1.1", prec);
    arb_printd(a, 40);
    printf("  FINE \n");

    arb_set_d(a, 1);
    arb_printd(a, 40);
    printf("  FINE \n");

    arb_clear(a);
    return 0;
}

I obtain:

1.100000000000000088817841970012523233891 +/- 0  WRONG 
1.100000000000000000000000000000000000000 +/- 1.7272e-77  FINE 
1.000000000000000000000000000000000000000 +/- 0  FINE

This clearly shows that the issue originates from the "arb_det_d" function when fed with a non-integer value. I will ask about this on the FLINT GitHub issue list.

This is not an answer, but a possible clue that it is a problem of ARB and not the SageMath wrapper.

When I run this code in my IDE:

#include "flint/arb.h"

int main(int argc, char **argv)
{
    arb_t a;
    arb_init(a);

    slong prec;
    prec = 256;

    arb_set_d(a, 1.1);
    arb_printd(a, 40);
    printf("  WRONG \n");

    arb_set_str(a, "1.1", prec);
    arb_printd(a, 40);
    printf("  FINE \n");

    arb_set_d(a, 1);
    arb_printd(a, 40);
    printf("  FINE \n");

    arb_clear(a);
    return 0;
}

I obtain:

1.100000000000000088817841970012523233891 +/- 0  WRONG 
1.100000000000000000000000000000000000000 +/- 1.7272e-77  FINE 
1.000000000000000000000000000000000000000 +/- 0  FINE

This clearly shows that the issue originates from the "arb_det_d" function when fed with a non-integer value. I will ask about this on the FLINT GitHub issue list.