The scenario you mention doesn't necesserarily requires to have Philip die in the womb of his mother. If he is born a girl or dies childless, the situation becomes pretty much the same.
As much as it seems possible, I'm not sure Henry II would be interested in the conquest of France. For one, he has no inheritance rights on it. Then, there is the mess this situation would represent.
The possible candidates to the French throne in this scenario would be the following ones :
Henri I of Champagne (1127-1181) - Count of Champagne and husband of Marie of France, eldest daughter of Louis VII. He had four children with his wife, including Henri II of Champagne (who later became King of Jerusalem) and Theobald III of Champagne.
Theobald V of Blois (1130-1191) - Count of Blois and brother of the former. Husband of Alix of France, second daughter of Louis VII. With her, he had seven children including his successor, Louis of Blois.
Henry the Young King (1155-1183) - Eldest son of Henry II Plantagenêt and co-King of England. Husband of Margaret of France, third daughter of Louis VII. The couple had a stillborn son (William) and the difficult pregnancy seems to have rendered Margaret sterile (she didn't had any children from Henry after the death of her stillborn, neither did she have one from her second husband Bela III of Hungary).
Richard the Lionheart (1157-1199) - Second son of Henry II Plantagenêt and in charge of the Duchy of Normandy. He is fianced to Alix de France, fourth daughter of Louis VII, but his claim would only be strong if he married her.
Robert I of Dreux (1125-1188) - Younger brother of Louis VII and his closest male relative. Salic Law would make him heir, but it wasn't officially applied before 1316 OTL, so it doesn't make him the only heir available. From three marriages, he had twelve children, including his son and heir Robert II.
This makes us with three sides in the possible Successoral War :
-The House of Blois-Champagne : The Blois-Champagne were extremly powerful at that time. With the eldest branch being Counts of Champagne and the cadet being Counts of Blois, they were encircling the Royal Dosmaine. Plus, Louis VII's third wife, Adèle, was from that family. They could also count on Henry White Hands, Archbishop of Rheims.
-The Angevins/House of Plantagenêt : Of the three sides, they have the lowest claim but are the strongest in France since they hold the whole Western half (Normandy, Anjou, Maine, Tourraine, Aquitaine and Britanny). Problem is that the family is far from United.
-The Remaining Capetians : This includes Robert I of Dreux, but also the last brother of Louis VII, Peter of Courtenay. This also includes Hugh III of Burgundy, who is a distant cousin but could support them for family reasons. One of Robert of Dreux's sons is also Bishop of Orléans, making him one of the twelves peers of France.
Two of the six Ecclesiastical peers could be supporting one side because of Family as I mentionned in the description. I don't really know how the Four remaining would play their part.
It leaves us with the six laïc peers, but four of them (Duke of Normandy, Duke of Aquitaine, Count of Champagne, Duke of Burgundy) are already linked to one of the sides. This leaves us only the following two :
Philip of Alsace, Count of Flanders - His policy were generally playing on a balance between the French and English Kings. I think he is pretty much a wild card in that scenario.
Raymond V, Count of Toulouse - Forced Vassal of Henry II, he is the husband of Constance, sister of Louis VII and Robert I of Dreux. Doubt he would support the Angevins, but don't know if he could support the Capetians : his situation is pretty bad at that point.
With the situation being the one I described, I'm not sure Henry II would be willing to push for one of his son to be the French King. That and there is also the fact it could be kind of risky : Henry's sons (except for John who was Henry's favorite) were very rebellous. Giving them the French crown could give them more power and would be risky.
I'm not saying this id impossible. Once I have finished my Napolenic Timeline (which I should update...), I was thinking on making a timeline working on that scenario (well, something close at least). But, I just pointed out the main difficulties.