In In the Garden of Beasts, what was Dodd's reason for inviting his two adult children to accompany him?
The answer to this question can be found on page 22 of the paperback edition that is searchable on Amazon. My own copy is on Kindle so I cannot independently verify that page number.
On the preceding pages, Erik Larson tells us that Dodd was initially somewhat reluctant to become the ambassador to Germany. When Franklin D. Roosevelt had been elected as president, Dodd had hoped to be an ambassador to a much less important country such as Belgium or the Netherlands. He was not sure he was cut out for being an ambassador to a big, important country and one that was in the midst of such upheaval.
Dodd ended up accepting the job for familial reasons as much as anything else. Larson says Dodd felt as if he had failed his wife by remaining as an obscure nobody his whole life. He therefore thought that he should make it up to her as much as possible by taking an important position now that it was offered.
Similarly, he invited the children along for familial reasons. Larson says that Dodd was feeling nostalgic because of the fact that he was getting old and his children were growing up. He wanted to have a few more years in which they could be together as a family. Larson quotes Dodd as writing to his daughter
I can not endure the thought of our lives all going in different direction—and so few years remaining.
In short, then, Dodd invited his children because he missed having his family together. He felt that he was getting old and would die before too long. Therefore, he wanted to have a bit more time that could be spent with his entire family.