Thank you, @Reeza. In order to keep the ANOVA and parameter estimates that I'd get from PROC REG, I'm thinking I'll just piece the output from PROC REG and PROC SGPLOT together.
data have;
input id weight_1 weight_2;
cards;
1 80 81
2 150 155
3 190 170
4 201 203
5 204 211
6 180 188
7 170 171
8 177 169
9 90 82
10 99 101
;
run;
ods trace on;
proc reg data=have;
model weight_1 = weight_2;
ods exclude diagnosticspanel;
ods exclude residualplot;
ods exclude fitplot;
run;
proc sgplot data=have;
title "Fit Plot for weight_1";
reg y=weight_1 x=weight_2 / cli clm;
lineparm x=0 y=0 slope=1 / lineattrs=(color=red);
xaxis values=(50 to 250 by 50);
yaxis values=(50 to 250 by 50);
run;