You have a horse?! I'm so jealous! (But not of the getting up early every day, mucking out in the freezing cold or the expense!)
That does make pregnancy doubly terrifying though! As well as missing riding (man that does wonders for my head!) you must be worried about what to do with him or how you'll cope with him after having the baby. (There are ways and means!)
I've known of people continue riding for some time - if he's safe to do some plodding, and you're happy to, there's no reason not to. Could you also walk him in-hand, lunge or play in the school with him? Even ride with someone, not leading exactly, but on foot just in case. To maintain the contact with your 'first baby' and keep you moving around outside?
Of course, if he's at all boisterous or unpredictable this might not be safe. Have you got a sharer/part loan to help with the heavy work? There must be tons of people willing to muck out for free riding (I would). Mucking out won't be helping your back, and tiredness won't be helping your mood.
Deffo insist on your risk assessment ASAP. Your employer is obliged to make reasonable changes to keep you comfortable - a new chair, and/or exercise ball, frequent breaks to move around or rest, a foot rest to help with your positioning.
This must be a real change to your normal self & routine. I can understand how you must be feeling. And hormonal is normal! 