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The cited source above claims things in a two columns paper, that cannot be valid.

To support this, and to give an answer to the question "how can I modify my work?", please let us consider an other source using:

the same hyperelliptic curve
over the same prime
where there appears a divisor with Mumford representation $(u,v)$ with more "realistic" coefficients in $u,v$:

http://shodhganga.inflibnet.ac.in/bitstream/10603/102010/15/15_chapter%205.pdf

Let us take a look at Table 5.4. inside the above Chapter 5, and adapt the code to work with the new objects in this working reference:

p = 4112543547855339322343814790708185367671872426434747235319998473455582535888229747778325047393413053
K = GF(p)
R.<u> = K[]
f = u^5 + 7943193*u^4 + 6521255*u^3 + 1065528*u^2 + 3279922*u + 3728927

C = HyperellipticCurve( f )
J = C.jacobian()
X = J(K)

U = ( u^2
      + 1860389956661272000673008332624408105124473591357802495195699152127038244919646073105431091649161433*u
      + 3678638643325468767033409217184617082278646180398696096065435808423595790408365381053077562453250122 )

V = ( 1442288836874733903146967744806211994245652783209857240169344848296298432525255771657442734321755066*u
      + 921980454689397557939882885330855476778834577885555086747866424426336117619472682642574975583793355 )

D = X( [ U,V ] )

key = 11794224706464405453771880923682985303502368223256244790322242497664987902757947838816724010526943231
div = key * D

And sage confirms the printed result from this Chapter 5, page 107:

sage: print div
(x^2 + 1334417218673579196183082084171449472807365544624463016735162813781624145553691422830410095888029556*x + 3947000779241763670672412101498061083590545643436596730239170524313050659358606510365451650167089110, y + 3685255333461212784675197538912907368200372565862892461204210868447776269393357974512751786077530346*x + 4039903126531302725487748758957892552553012169705313271281907787620627040317525523860359172450156937)

Note: The computation also gives the warning:

verbose 0 (3324: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
verbose 0 (1083: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.

Note: This post is related to 40509 on this site.

Note: As mentioned in the comment, the coefficients of $(u,v)$ are expected to be (at least for $k\cdot D$) as "big" as the prime $p$...