You should use a chance-corrected index of categorical agreement with ratio weights; I would recommend the generalized form of Bennett et al.'s S score. The formula would be:
$$S=\frac{p_o-p_c}{1-p_c}$$
$$p_o=\frac{1}{n'}\sum_{i=1}^{n'}\sum_{k=1}^q\frac{r_{ik}(r_{ik}^\star-1)}{r_i(r_i-1)}$$
$$p_c=\frac{1}{q^2}\sum_{k,l}w_{kl}$$
$$r_{ik}^\star=\sum_{l=1}^qw_{kl}r_{il}$$
$$w_{kl}=1-\frac{[(x_k-x_l)/(x_k+x_l)]^2}{[(x_\max-x_\min)/(x_\max+x_\min)]^2}$$
where $q$ is the total number of categories,
$w_{kl}$ is the weight associated with two raters assigning an item to categories $k$ and $l$,
$r_{il}$ is the number of raters that assigned item $i$ to category $l$,
$n'$ is the number of items that were coded by two or more raters,
$r_{ik}$ is the number of raters that assigned item $i$ to category $k$,
$r_i$ is the number of raters that assigned item $i$ to any category, and
$x_k$ is the actual numerical value of category $k$.
You can calculate this value automatically if you format the data correctly and use my mSSCORE function in MATLAB or Kilem Gwet's agree.coeff3.raw.r function in R (the bp.coeff.raw subfunction).
It is not necessary that all cases be rated by both raters, but it would be preferable to have as many as possible rated by both. Including more participants will yield a narrower confidence interval on the reliability estimate, making it more informative and trustworthy. You also need to be able to argue convincingly that the participants rated by a single rater are no different than those rated by multiple raters (and that therefore the known reliability in the latter is representative of the unknown reliability in the former). The best way to do this would be to randomly assign participants to be rated by one or both raters and to do this throughout the project rather than doing an initial reliability test and then assuming that that is the reliability that will continue forever. For more introductory information about observational measurement, please see my recent article on the topic.
