Arthur should be in the 1900 US census at about six years old. There’s only one BULLARD family in Brazoria County, TX, 1900:

1900-TX-Brazoria-Justice Precinct #7-ED 8-Sheet 2A

Bullard, Christopher Head BM May 1836 64 M40 TN TN TN
Eliza Wife BF Apr 1834 66 M40 13-2 NC NC NC
Harris, William G-Son Nov 1884 TX
Adeline G-Dau Jan 1887 12 TX
Bettie G-Dau BF 1892 7 TX

I searched for Arthur in 1920 in Brazoria, assuming Wharton is in Brazoria, with no luck. We could have a spelling issue (I tried BALLARD).

Hope this helps.
Preston