Oat milk is a suitable drink for Bryan but he might so be full of it he does not feel hungry so perhaps Elsie could try to offer him water instead. Iron-rich foods which Bryan could be offered include fortified breakfast cereals, dark green vegetables, like broccoli, pulses, such as baked beans, lentils and kidney beans, dried fruit, such as dried apricots, figs, sultanas and prunes, and ground nuts or nut butters.
In order to absorb the iron, food containing vitamin C must be taken at the same time as the iron-rich food. Good sources of vitamin C include oranges, strawberries, raspberries, blackcurrants and kiwi fruit, tomatoes and red or green peppers, dark green, leafy vegetables, potatoes and sweet potatoes.
It is quite normal for toddlers to go through phases of fussy eating. Things which Elsie could do to help Bryan include eating at the same time as him, introducing new foods gently, offering him some choice but not too much, offering food little and often. The British Nutrition Foundation offers some useful guidance about a balanced diet and portion size (https://www.nutrition.org.uk/healthyliving/toddlers.html)
It would be helpful to check and plot Bryan’s height and weight and to compare this to previous recordings in order to assess his growth. In view of the restricted diet and Elsie’s concerns it might be best to refer Bryan to a dietician for specialist assessment and advice.