VBIC Mextram Transistor Spice Models Parametrization
I did some BJT modelling previously, from which someone posted a link to the full document on the MEXTRAM model, and also the VBIC model paper may help you.

Here is my condensed advice: don't use the default models from LTspice, write your own .txt file that explicitly defines the parameters that you want.

Writing your own LTspice model

For example when I was trying to model a germanium BJT I defined a text file called bjtmod.txt, which contained only one line

.model IDEAL_GERM_NPN NPN(IS=1E-6, BF=120, BR=5)


You can see here first the statement is to define a .model, called IDEAL_GERM_NPN, which is an NPN BJT with parameters IS=1E-6, BF=120, BR=5.

To use this in LTspice you must place it in the same folder as the schematic you are working on. Including it to the schematic is done by writing a SPICE statement:

.include "bjtmod.txt"


To set your BJT to use this model make sure that the Value field is set to IDEAL_GERM_NPN.

You can then customize each parameter of the model, including avalanche terms. From the reference you provided:

.model 2n3904L4 NPN(level=4, rcx = 2.65711, rci = 0.0, rbx = 0.634251, rbi = 5.203349, re = 0.0001, is = 1.26532e-10, nf = 1.5, nr = 2.89609, fc = 0.512134, cje = 4.64214e-12, pe = 0.4, me = 0.256227, cjc = 3.015688e-12, cjep = 7.53922e-13, pc   = 0.4, mc = 0.238109, cjcp = 0, ps   = 0.75, ms   = 0.5, ibei = 6.13333850374693e-13, nei = 1.5, iben = 2.30771e-09, nen  = 3.31052, ibci = 6.13333850374693e-12, nci = 2.89609, ibcn = 2.30771e-09, ncn = 1.9876, vef  = 566.483710419377, ver  = 8.69028854282359, ikf = 0.0272221, ikr  = 0.272221, tf   = 4.19578e-10, xtf  = 0.906167, vtf = 8.75418, itf  = 0.0105823, tr  = 6.82023e-08, td   = 0, ea = 1.05, eaie = 1.05, eaic = 1.05, eane = 1.05, eanc = 1.05, xis  = 1, xii = 0.9, xin = 0.668948, kfn  = 0, afn  = 1, avc1 = 0.02, avc2 = 110)


Lastly, don't get fixated on whether it resembles a real BJT. It won't. Some of these parameters vary hugely between devices. Extraction procedures are phenomenally complex, take that from someone who wrote one for only around 12 BJT parameters.