Probably due to the fact that Male Avatar saved Chrom's Exalted arse many times, and considering Luce's other marriage options such as an unfaithful philander and a bitter masked man Im sure Chrom is fine with it.
I'm going to consider that comment about baby Lucina a joke in poor taste. XP
As for Chrom letting you marry his daughter, why not? Some people may be uncomfortable with their best friends getting in relationship with family members, but not everyone would. Chrom completely trusts the Avatar, so it's understandable that he wouldn't mind his time-shifted daughter marrying his best friend. Or said friend marrying his sister either of them or son. I mean, they're around the same age, for crying out loud. Sure there's the time traveling business, but mentally and physically, they are around the same age.
And this is coming from someone who would rather be Lucina's mother than lover. :P --- "What good is having friends if they don't put you in mortal peril every once in a while?" - Melody Muns, The Rithmatist
http://safebooru.org/index.php?page=post&s=view&id=865258 --- "A little rudeness and disrespect can elevate a meaningless interaction to a battle of wills and add drama to an otherwise dull day." - Bill Watterson
Because for any point of logic, he's in favor of it-- obviously if that's the case, he's -not- going to offer resistance. He's not dumb-- he knows that Timetraveler!Lucina is of an age where she's likely going to fall in love. And as a father, would you not want your daughter to be with someone who you trust? [Don't forget that Chrom trusts Robin with his very life.]
Flip flop- put yourself in Chrom's shoes: Your best friend (who, I'm going to point out is, someone you literally trust with your very life) is going out with your timetraveling daughter. Would you seriously be opposed to that- knowing full well how your friend acts (you already know he's the type who'd treat her right and put her happiness and well being first) and be in favor of her winding up with say someone who literally doesn't care about her goals in life, wouldn't care if you died in a ditch somewhere,etc.; the resident playboy; her (also timetraveling) cousin; or some other guy who you have no -real- clue is going to treat her; simply because he's your friend?
If you expand upon that, Robin taking Lucina does mean she's not winding up with someone like the mentioned. (Yes, I'm being cruel with my words towards Gerome, but he... sorta... does exactly that. He is a complete defeatist in everything- and he's an extremely bad match with Lucina. (Yes, they're both serious, but they're also opposite in ways that just wouldn't work. I cannot possibly see them being together as anything but one of those miserable marriages that wind up divorcing each other within a year or so (and then Lucy's alone. What is an alone Lucy going to do? Pull an Ike.)). (He's also a male Tsundere, which why I'm think he's such a good match with Severa: Tsun-Tsun, Dere-Dere.)) Chrom would know Lucina is going to be able to be happy with Robin. It's pretty obvious why he'd be in favor of it- he's old enough to understand that the Timetraveling Lucina will need someone else in her life- and who better for her than his best friend?
As far as Baby!Lucina goes? Her and TT!Lucina are two different people. Enough said. --- http://i.imgur.com/vSQXpF4.jpg (Cutest Giant Robot ever) http://i.imgur.com/rc9CO0c.jpg
What I want to know is where the hell Baby Lucina is while Chrom and his wife are off taking on Walhart and Grima's forces.
Had Robin marry Chrom, and the two of them were in basically every single cutscene from that point on but... baby lucina was nowhere to be seen! Was she left all alone in Chrom's tent? Did they forget her back in Ylisse? Did she fall through a plothole? Poor child, growing up without her parents paying any attention to her.