Well, there's a really simple answer to this: While US-Soviet relations were always, in the best of times, frosty, they were still a global superpower on the world stage and to send some minor dignitary would've been perceived by the Soviets (as well as the rest of the world) as a massive and intentional snub and would only have exacerbated the problems between us. Contrary to his generally hostile rhetoric, even Reagan wasn't foolish enough to want to deliberately push the two superpowers into a further state of conflict. In fact, Reagan generally followed the doctrine of relations established since the Truman Administration. That was essentially to spend them into defeat. The Soviets did not have the same vast array of allies and trading partners to do business with, nor did they have the natural resources to match the U.S. in a cold war of attrition. Defending their interests abroad was ultimately costlier for them than it was for us.

Getting back to your question, the simple answer really was it was diplomatic precedent for the two superpowers to send their number two to the funerals of the leaders. For example, the Soviets sent Anastas Mikoyan, who was First Deputy Chairman of the Council of Ministers (Essentially the Deputy Prime Minister) to JFK's funeral. So, this was the long established diplomatic order. We're not going to have the President go because we're not friends, but we're not going to snub you by sending nobody since you are the other global superpower.